The announcement comes amid concerns about the spread of the Delta variant, which now accounts for 99% of all confirmed and probable cases of coronavirus across the UK.
While travel rules have been tightened for those coming from the UK, the Italian minister said rules for travellers from other European Union countries, the United States, Canada and Japan would be relaxed.
Mr Speranza announced on Friday that he signed a new order allowing entry for travellers from those countries with Green Certificate requirements - if visitors show they have been vaccinated or recently tested negative for Covid.
The travel ban for people coming from India, Bangladesh and Sri Lanka will remain in place, he added.
The minister did not say when the measures would be introduced.
There were similar moves by France, Austria and Germany to tighten rules on arrivals from Britain.
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Public Health England (PHE) has pinpointed 33,630 more cases of the variant in the UK compared to last week - up by 79% on the previous week - although an increase in testing is likely to have contributed to the rise.
The data shows that 75,953 confirmed and probable cases of the Covid-19 Delta variant have now been found in the UK.
Of the 75,953, some 70,856 have been in England, 4,659 in Scotland, 254 in Northern Ireland and 184 in Wales.
Hospital cases have also risen, though most of those needing treatment have not had a vaccine.
In the last 24 hours, the UK has reported 10,476 more Covid infections and 11 more deaths.
Stay up to date with the changing travel situation in the UK and abroad with news, information and advice at itv.com/holidaynews.