As of last year, new rules prevent politicians from sitting at both Westminster and Holyrood at the same time.
Following the Holyrood election defeat Sarwar promised to stay on as leader in Scotland.
In May, as reported by The Herald, external, when asked if he would ever take up a seat in the House of Lords, Sarwar said: “No. I’m focused on being first minister of this great country.”
He added: “My only commitment, my only focus, is serving my constituents in Glasgow and serving the great people of this country, if I get the honour of being first minister.”
After Sir Keir’s resignation in June, when asked if he planned to serve as party leader for the full five-year Holyrood term he said: “That is my intention.
“I’m hungry for the fight ahead and I want to play my full part in making sure this country never ends up in the hands of Nigel Farage and Reform.”
Earlier, Scotland’s First Minister and SNP leader John Swinney wished Sarwar well in his “exciting new job” but he was critical of Burnham.
Speaking to Radio Scotland’s Breakfast programme, Swinney said: “It is a terrible look for Andy Burnham because the new PM has made a great play of essentially improving people’s perceptions of politics and politicians.
“And to make one of his first decisions a decision to take the Labour leader in Scotland, who has just led the Labour Party to their worst result in the Scottish Parliament elections in the history of the Scottish parliament elections, and give him a job for life in the House of Lords and a job in the UK government is in my view a very, very strange way of trying to rebuild the confidence of the public in politics.”