"We need a proper competition with the best candidates in the field, and I will run," he said, according to Sky News, during an appearance in London on Saturday.
Streeting says he does not want to force an immediate leadership election to replace the embattled Starmer, as it is not in the "interests of the party or the nation".
The issue is that Streeting wants to postpone the leadership election to, for example, give a possible candidate like Andy Burnham the chance to participate. Burnham is the mayor of Manchester and does not sit in Parliament - a formal requirement to become leader.
A hasty election would risk the legitimacy of Starmer's replacement and lead to continued "instability and uncertainty," according to Streeting:
"If you think you are good enough to do the top job, if you think you have the right ideas for the country, you should be able to put them forward and test them against other contenders," he said.
When 43-year-old Wes Streeting spoke on Saturday at the Labour-linked think tank Progress, he called Britain's exit from the EU a "catastrophic mistake" and called for rejoining, writes Sky News.
"Britain's future lies in Europe - and one day back with the European Union."
Demands for Keir Starmer to resign have recently accelerated after the social-democratic Labour Party's major defeat in regional and local elections.
