Allies of Wes Streeting have said he is preparing to stand down as health secretary amid deep frustration with Keir Starmer’s leadership, and could mount a formal challenge for the leadership as early as Thursday.

Downing Street insiders had suggested Streeting did not yet have the required support from 81 MPs, which is needed to formally launch a leadership bid, after the prime minister issued a “put up or shut up” ultimatum to his cabinet.

A source close to the health secretary told the Guardian on Wednesday that he was planning to resign on Thursday and launch a leadership bid. The idea that Starmer had seen off a putsch was “laughable”, they added.

“No-one has the numbers till the bell is struck, even canvassing isn’t real, people need certainty before they write their name down. But he thinks he’s got the numbers,” they said.

A second MP, also close to the Streeting camp, said they had been involved in discussions about getting the requisite numbers he would need to trigger a contest, and had begun ringing round MPs.

Two other MPs said they had been called by allies of Streeting on Tuesday evening to tell them: “He’s going for it.” One said they were unsure if the health secretary had sufficient backing to go through with the plan.

Streeting held brief talks with Starmer in Downing Street on Wednesday morning, but the health secretary’s spokesperson said he was not planning to comment afterwards in order not to distract from the king’s speech.

Another MP, a close ally of Starmer, said the prime minister had always believed Streeting would not win the party and that he would fight any challenge – despite intense pressure from MPs who have publicly called for him to go.

One cabinet ally of the prime minister told the Guardian: “Personally I’d throw him in the river but luckily Keir Starmer is a calm and patient person.”

Earlier on Wednesday, Zubir Ahmed, a close ally of Streeting who quit as a minister on Tuesday, renewed calls for the prime minister to resign, saying his authority had “irretrievably ebbed away”.