In a news conference later on Friday, Brown said he had no plans to step aside as prime minister and said that he would introduce a regulatory body to supervise politicians' expenses.
"I am, therefore, calling on you to stand aside to give our party a fighting chance of winning"
James Purnell MP, in open letter to prime minister
Alan Fisher, Al Jazeera's correspondent in London, said the prime minister's decision not to resign was not a surprise given how much Brown had craved the job in the first place.
Fisher said Brown faced an uphill battle to win the next election and that "he has a lot of work to do to reconnect with the public".
Early results from Thursday's local authority elections showed signs of a swing against the ruling Labour party, with it losing seats to the Conservatives and Liberal Democrats.
Labour members of parliament have also been gathering signatures to unseat Brown, increasing the possibility of early elections in the country.
Purnell, who resigned late on Thursday, called in a letter to the Times newspaper for Brown to go.
"I now believe your continued leadership makes a Conservative victory more, not less, likely," he wrote.
"I am, therefore, calling on you to stand aside to give our party a fighting chance of winning."
Brown's government and the country's major political parties have been hit hard by weeks of leaked details about politicians' expense claims.
Darling has repaid some money after he acknowledged mistakes in his expenses claims and has faced criticism over his manipulation of the housing allowance system.
Jacqui Smith, then interior minister, was the first to announce that she was quitting over the scandal. Hazel Blears, another minister, followed suit.
Johnson, tipped as the likeliest replacement for Brown, has urged colleagues to unite behind their leader in the wake of likely poor election results.
"I continue to believe that Gordon Brown is the best man for the job," Johnson said in a statement.
"It is vital now, more than ever, that we unite for the sake of the party and the government."
Brown replaced Tony Blair in June 2007. He has gained praise overseas for his handling of the global economic crisis, but seen his domestic approval ratings tumble.