Meanwhile, Keir Starmer, seeking to secure a legacy before he leaves office, faces the weekly Prime Minister's Questions session in parliament before flying to Berlin to meet European allies for talks on Ukraine and the Middle East.
Starmer announced his plan to resign on Monday and will be out of office within weeks once the governing Labour Party picks a new leader.
Jones, a Starmer ally, had been encouraged to run so that Burnham faces a test of his ideas and policies in front of Labour MPs and members.
Others argue that a leadership contest will only focus attention on the party's internal divisions and extend a period of political uncertainty.
Jones told Sky News that running for the leadership is "not something that I'm going to do".
But he cautioned Burnham against veering too far to the left in economic policy, a concern of some in the business and financial worlds.
Burnham is expected to choose a new treasury chief to replace Starmer appointee Rachel Reeves.
Jones said it must be someone "that can reassure the markets, reassure the trade unions and reassure the parliamentary Labour Party, and by extension the public".
Burnham is expected to make a speech within days outlining some of his economic plans.
Starmer is leaving after two years in office marred by missteps and judgment errors that eroded his standing with his party and the public.
Burnham, a former cabinet minister who served since 2017 as mayor of Greater Manchester, won a special election on June 18 for a seat in parliament with the express aim of challenging Starmer for leadership of the Labour Party and the country.
So far, he faces no challengers.
Former health secretary Wes Streeting, who was considered his main rival, says he will back Burnham.
Nominations for the Labour leadership will open on July 9 and close a week later.
If Burnham is the only contender, he could be prime minister by July 17.
Starmer told a cabinet meeting on Tuesday that he would try to oversee an "orderly transition" to his successor.
He is also keeping up a busy schedule, trying to cement a legacy for his shortened term in office.
Starmer's trip to Berlin for a meeting of the "E5" - Germany, France, Italy, Poland and the UK - for talks on European defence, the war in Ukraine and conflict in the Middle East, underscores the role he has played on the world stage.
The British government is expected to publish a long-awaited defence investment plan - which sparked the resignation of defence secretary John Healey on June 11 - before a NATO summit in Turkey on July 7 and 8 that Starmer is likely to attend.
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