The Prime Minister secretly discussed the plan with Mr Robertson when they both attended the Palio horse racing event in Siena last week during Mr Blair's Tuscan holiday.
A decision could be made next week but senior Labour Party sources said there were advantages in calling an early by-election. "We think we can win it at any time against the SNP but it would enable us to move before they had time to prepare," said one source.
Mr Robertson is standing down to take up his appointment as secretary general of Nato at the end of September.
Taking a peerage would be one of the quickest ways to trigger a by-election, and party sources said it would enable Tony Blair to keep Mr Robertson's "talents and expertise" at Westminster for the long term.
"Most of the discussions are over the telephone but they did meet at Palio," said the source. "It could go either way at the moment." The other option would be to delay the by-election until the Commons returns from its summer recess in October. Taking a peerage could trigger the by-election for 23 September, coinciding with the start of the SNP conference, when its party workers would be occupied elsewhere.
The by-election will be the first big contest for Labour since the opening of the Scottish Parliament and the SNP is determined to make an impact. The nationalists scored a historic victory in the constituency - then named Hamilton - in 1967 when Winnie Ewing took the seat in a by- election. This gave the party its first Commons representation since the Second World War but Labour regained the seat in the 1970 general election. Mr Robertson won the seat in a 1978 by-election when he doubled Labour's majority to 6,492 on a 4.5 per cent swing.
The SNP deputy leader, John Swinney, said calling a ballot in the party conference season amounted to jiggery-pokery". He added: "The people of Hamilton South will not take kindly to a by-election timed only to give their representative an ermine robe before sending him off to Europe, and will resent being manipulated by New Labour in this way."
He accused Labour of taking the seat for granted, predicting that it was "likely to pay a heavy electoral price for such arrogance". He went on: "The SNP's preparations for the by-election are well advanced and we will put a formidable campaign in place to guarantee a strong and decisive contest that will give Labour the maximum pressure and the SNP the maximum opportunity to win."