Downing Street ordered a leak inquiry over the weekend after reports Boris Johnson was poised to unveil another lockdown.
A statement by the prime minister was also hurriedly brought forward to Saturday night.
Sir Lindsay said any MP found to have leaked details about the second lockdown should apologise for displaying "discourteous and unacceptable" behaviour.
He also told Mr Johnson he expected MPs to be kept updated about the inquiry.
In a statement, Sir Lindsay said: "After speaking with the Prime Minister and the Leader of the House (Jacob Rees-Mogg), who went to great lengths to assure me that the leaks were not from Downing Street, I expect the Prime Minister to keep the House updated on his leak inquiry.
"I also hope that if the leaker is identified and if a member of this House, that member will make a full apology to the House for the discourteous and unacceptable behaviour."
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies