The Prime Minister had come under sustained pressure to say whether Mr Crosby, whose lobbying company has worked for tobacco giant Philip Morris, had spoken to him about shelving the plan for cigarettes to be sold in plain packaging.
On Tuesday, Mr Crosby denied the claims as "simply false." The Cabinet Secretary has also dismissed Labour calls for an inquiry.
But Mr Osborne refused to answer repeated questions from Economics Editor Richard Edgar over when the "principles of engagement" document was agreed:
Cabinet Secretary Sir Jeremy Heywood told Labour he does "not see what purpose would be served" by an inquiry into claims that Tory adviser Lynton Crosby lobbied David Cameron over tobacco packaging.
In a letter to Labour leader Ed Miliband, who has demanded an inquiry, Sir Jeremy said:
Mr Crosby is not employed by the Government. He is contracted by the Conservative Party.
He has not been granted permission to have access to Government papers or attend Government meetings. Nor does he advise or lobby on Government business, such as the regulation of cigarette packaging or fracking.
His role, as I understand it, is to advise the Conservative Party and any meetings he may have had with the Prime Minister are on this basis, which is not in itself a conflict with the ministerial code.
Cabinet Secretary Sir Jeremy Heywood has rejected Labour calls for an inquiry into alleged breaches of the ministerial code relating to lobbyist Lynton Crosby's engagement as elections adviser to the Conservative Party.
Labour vice-chairman Michael Dugher said Lynton Crosby's statement denying he lobbied the Government on tobacco issues "raises more questions than it answers":
This baffling statement raises more questions than it answers. David Cameron has refused to deny that he has had a conversation with Lynton Crosby about tobacco policy on at least 16 occasions. If Lynton Crosby is telling the truth, why on earth couldn't David Cameron say this himself?
The fact remains that David Cameron chose to bring a tobacco lobbyist into the heart of his Government, changed his policy on cigarette packaging and was then unable to give a straight answer about Lynton Crosby's influence.
Conservative election adviser Lynton Crosby has denied having "any conversation or discussion with or lobbied the Prime Minister, or indeed the Health Secretary or the health minister, on plain packaging or tobacco issues".
The Prime Minister will not be able to move on from the Lobbying row until he answers questions over Lynton Crosby's influence on government policy, according to a public relations expert.
Kevin Craig, from Political Lobbying and Media Relations, said Lyton Crosby's involvement in government policy "looks wrong and and doesn't feel right" and called on the Prime Minister to provide answers.
The NHS links pre-dated David Cameron hiring Mr Crosby however it is the second time the Prime Minister has had to defend his controversial election strategist over claims of lobbying.
Mr Cameron has insisted that Mr Crosby had not intervened in policy decisions, but has repeatedly refused to answer questions about the extent of his conversations with the lobbyist.
The Prime Minister has come under sustained pressure to say whether Mr Crosby, whose lobbying company has also worked for tobacco giant Philip Morris, had spoken to him about shelving the plan for cigarettes to be sold in plain packaging.