Video report from ITV News Political Correspondent Romilly Weeks
The joint statement came after the commission said it would tighten the EU’s vaccine export rules, prompting fears it could bar supplies being sent to the UK.
Earlier on Wednesday, Boris Johnson also warned that a trade war over jabs would result in “considerable” and “long-term” damage.
In a joint statement, the UK government and European Commission said: “We are all facing the same pandemic and the third wave makes cooperation between the EU and UK even more important.“We have been discussing what more we can do to ensure a reciprocally beneficial relationship between the UK and EU on Covid-19.
“Given our interdependencies, we are working on specific steps we can take - in the short-, medium- and long term - to create a win-win situation and expand vaccine supply for all our citizens. “In the end, openness and global cooperation of all countries will be key to finally overcome this pandemic and ensure better preparation for meeting future challenges.“We will continue our discussions.”
It came after the European Commission set out a tougher regime to stem supplies of jabs to nations faring better in the pandemic as the bloc’s states faced a third wave of cases.
Admitting it is a Covid-19 “hotspot”, the European Commission said on Wednesday it may not approve exports to nations with more advanced vaccine rollouts or where there is a better “epidemiological situation”.
The union said 10 million doses have been exported from the bloc to the UK, while none have been imported in the other direction.
European Commission president Ursula von der Leyen said: “While our member states are facing the third wave of the pandemic and not every company is delivering on its contract, the EU is the only major OECD producer that continues to export vaccines at large scale to dozens of countries.
“But open roads should run in both directions.”
The EU said it will consider two key factors before authorising vaccine exports.
First, they will consider "reciprocity": whether the destination countries restricts its own exports of vaccines.
Secondly, under “proportionality”, they will consider whether the epidemiological situation in the destination country is “better or worse than the EU’s”. This could include looking at the country's vaccination rate and access to vaccines.
But answering questions from senior MPs on Wednesday, the prime minister said he did not "think that blockades of either vaccines or of ingredients for vaccines are sensible, and I think that the long-term damage done by blockades can be very considerable.
“I would just gently point out to anybody considering a blockade or an interruption of supply chains that companies may look at such actions and draw conclusions about whether or not it is sensible to make future investments in countries where arbitrary blockades are imposed.”
ITV News Europe Editor James Mates gives his analysis from Brussels
Also on Wednesday, European Commission executive vice-president Valdis Dombrovskis insisted advance purchase agreements would be respected.
He told reporters in Brussels the EU faces a “very serious epidemiological situation” but “continues to export significantly to countries whose epidemiological situation is less serious than ours or whose vaccination rollout is more advanced than ours”.
He added British-Swedish vaccine developer AstraZeneca has “only delivered a small portion of its agreed contractual commitments” with the EU.
He said the EU has exported at least 43 million doses of coronavirus vaccines to 33 countries since the end of January.
But “continued shortfalls in production are not distributed fairly across different contracting countries”, he said, in an apparent reference to the production of AstraZeneca jabs in the UK.
The EU “continues to export vaccines to countries that have production capacities of their own but when these countries do not export to the EU there is no reciprocity”, he continued.
The row over the supply of AstraZeneca jabs began in January after the company said it would not be able to deliver as many jabs in the first quarter of the year as originally promised. It said it would prioritise supply to the UK as the deal with the UK came before the EU one.
Mr Dombrovskis clarified the export authorisation mechanism is not addressed at any specific country and European Commission health commissioner Stella Kyriakides denied it was an "export ban".
Asked whether the Pfizer/BioNTech vaccine, the main vaccine exported from the EU to the UK, would be covered by the new rules, Mr Dombrovskis said: “Concrete decisions will be taken on a case-by-case basis.
“These adjustments to the regulation which we are presenting today do not describe a specific situation or a specific decision which is to be taken by member states and confirmed or objected (to) by the European Commission."
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Setting out the "alarming" situation in some EU countries, health commissioner Stella Kyriakides said 19 countries are reporting a rise in cases, 15 are reporting increased hospital and ICU admissions while eight are reporting a rise in deaths.
European Commission president Ursula von der Leyen first told reporters on March 17 the EU was seeking to block exports of coronavirus vaccines to countries with higher vaccination rates.
She ramped up the rhetoric this weekend, saying the EU has the power to "forbid" exports.
EU leaders are due to discuss the vaccine row during a virtual European Council summit on Thursday.
Across the EU, just over 11% of adults have received a first dose of a Covid-19 vaccine but in the UK the figure is more than 54%.
Concerns over UK supplies will centre largely on Pfizer, the main vaccine export from the bloc which is being produced in Germany and Belgium.
A spokesperson for Pfizer said it was assessing the “full implications” of the commission’s move, adding: “We have been clear and consistent with all stakeholders that the free movement of goods and supply across borders is absolutely critical to Pfizer and the patients we serve, particularly during this devastating global pandemic.
“Pfizer is deeply concerned by any legislation that threatens our ability to manufacture in, or export from, the European Union.”