Downing Street has said it would be "unacceptable and inappropriate" for any civil servant to claim Jeremy Corbyn was "too frail" to become PM.
No 10 said the cabinet secretary would write to the Labour leader after the party demanded an inquiry into alleged comments by officials to the Times newspaper.
It is unclear whether Sir Mark Sedwill will order an investigation.
Mr Corbyn has said the civil service has a duty to be non-political.
The Times reported on Saturday that it had been told by two senior civil servants that the 70-year old may have to stand down due to health issues.
The article drew an angry response from Labour, which denounced the comments as a "scurrilous" attempt to undermine the party's efforts to gain power.
Asked about the row, Theresa May's official spokesman said: "Impartiality is one of the fundamental values of the Civil Service and underpins its ability to effectively serve the government of the day.
"It would clearly be inappropriate and unacceptable for comments of this sort to have been made or briefed to the press."
Raising the issue in the House of Commons, shadow minister Jon Trickett called on ministers to "root out the miscreants" in the civil service responsible for the "undemocratic and unconstitutional" comments.
But responding for the government, Work and Pensions Secretary Amber Rudd suggested his criticism was premature, given the cabinet secretary had yet to look into the matter.
She said she had "complete confidence in the fairness and independence of the civil service".
Commons Speaker John Bercow said the principle that the civil service was politically neutral was an "absolutely sacred" one.
He added that the Labour leader "looked perfectly healthy to me".
Mr Corbyn has called it "a farrago of nonsense" and "tittle tattle".