With the backing of Britain and America as well as several other major European nations, Mr Robertson looks almost certain to get the job.
President Bill Clinton has already given the move his backing, and the US is believed to be keen to put a British figure at the head of the organisation. While the UK has had a close relationship recently with the US and has supported actions in Kosovo and Iraq, other European nations have been more sceptical.
If Mr Robertson is chosen to replace Javier Solana when he steps down from the job in the autumn he will have to leave Parliament, sparking a by-election in Hamilton South and a fresh cabinet reshuffle.
A statement from Downing Street last night said Mr Blair had received "a positive response" to inquiries on the subject at yesterday's Sarajevo summit from Jacques Chirac of France, Gerhard Schroder of Germany, Massimo D'Alemo of Italy and Jose-Maria Aznar of Spain.
"George has exactly the right mix of defence expertise and political and diplomatic skills. He would do an excellent job and we will be pressing his case very hard," Mr Blair said.
The Prime Minister's official spokesman added that Mr Robertson was "a reliable, heavyweight figure who has built a strong reputation through the Strategic Defence Review, which he led and which modernised our armed forces, as well as through the European Defence Initiative.
"In addition, several leaders Tony Blair spoke to pointed out that George Robertson had a good war in Kosovo, not least as a media performer," he said.
The nomination was revealed to other European nations' Nato ambassadors yesterday by the UK permanent representative, Sir John Goulden.
Mr Robertson said in a statement last night that it was a "great honour" to be invited to put his name forward.
"If I am appointed I will work to ensure that my experience in and out of government can be brought to bear for the good of the alliance as a whole."
He described the post of secretary-general as "one of the most important in today's world. "This is particularly so at a time when the alliance is committed to the full programme of action agreed at the Washington summit, as well as the international community's efforts to bring stability to the Balkans.
"As the alliance starts its second half-century it has to learn the lessons of its clear success in Kosovo, build the European Defence Identity and deal both with enlargement and relations with Russia and Ukraine. These are formidable challenges, which must be addressed for future generations and I would relish the chance to contribute.
"In the meantime I have an important job to do as Britain's Defence Secretary."
The nomination may well be a reward for his unstinting and enthusiastic support for Mr Blair's stance throughout the Kosovo conflict.
The news underlined reports that this week's decision by the Prime Minister not to reshuffle the Cabinet presaged a bigger reorganisation in the autumn. But the Prime Minister's spokesman warned the news should not spark a fresh round of the reshuffle speculation which has dominated the news in recent weeks.
"Before the media launches another wave of reshuffle hysteria, I tell them they will be making yet another mistake if they do so on the back of this. If George gets the job he can be replaced as Defence Secretary without too much fuss," he said.