Streeting, who has said he has the backing of the 81 Labour MPs needed to trigger a challenge, told BBC Newsnight late on Tuesday he thought a contest should be initiated sooner rather than later.
But Starmer, speaking at the G7 summit in Evian, repeated on Wednesday that he would fight to stay in his job if a formal challenge was launched against his leadership.
"If there is a challenge, I intend to fight in any challenge to my leadership," Starmer said.
"I don't think there should be a challenge, because I think that is a bad thing for the country."
The Labour Party is waiting to see if Andy Burnham, the Greater Manchester mayor, can win a local election in Makerfield on Thursday to return to parliament, where he would become the frontrunner in any challenge to Starmer.
Streeting, who quit in May in protest at the prime minister's record, said he wanted Starmer to take "time to reflect over the weekend and I think we should see where we are then".
"I would prefer the PM to take a decision on his own terms rather than leave it for me or Andy or anyone else to trigger a contest ... We can't carry on with this uncertainty and paralysis."
In a further blow to Starmer's authority, the highly respected defence minister, John Healey, resigned on June 11, accompanied by a scathing critique of Starmer's record in allocating funds to increase defence spending.
