The Cabinet Office will investigate whether senior officials told journalists they thought Jeremy Corbyn was “too frail” to be prime minister.
Theresa May’s spokesman said, if identified, those responsible would face “disciplinary action”.
However Labour sources dismissed the response as “inadequate” and called for a full independent inquiry.
Labour MP Laura Pidcock accused senior civil servants of trying to “sew doubt” about Mr Corbyn.
She said there was “a fear of the Labour agenda” among top mandarins who wanted to thwart a Labour government.
Over the weekend the Times said it was briefed by two senior civil servants with suggestions the Labour leader may have to stand down over supposed health issues.
Downing Street has previously said it would be “unacceptable and inappropriate” for any civil servant to make such claims and one ex-senior civil servant said it did “more harm to the civil service” than to Mr Corbyn.
Speaking to BBC Radio 5 live, Lord Kerslake said it would take time before trust in the service was “restored”.
Lord Kerslake – who advises Labour on preparing for government – said it was “absolutely essential the civil service maintains its impartiality” adding the leak had “set that back”.
“I think Jeremy Corbyn is rightly very unhappy about this story,” he said.