Former shadow home secretary Diane Abbott tweeted: "Crushing defeat for Labour in Hartlepool. Not possible to blame Jeremy Corbyn for this result. Labour won the seat twice under his leadership. Keir Starmer must think again about his strategy."
The left-wing Momentum group, which backed Mr Corbyn, said: "Starmer's strategy of isolating the left and replacing meaningful policy with empty buzzwords has comprehensively failed."
But Sir Keir's allies say the defeat, which saw a 16% swing from Labour to the Conservatives, means the party has not changed enough under his leadership.
Former Hartlepool MP Lord Mandelson said he hoped "very much that when Keir and his colleagues in the shadow cabinet say this means that we have got to change direction they actually mean it".
He told BBC Radio 4's Today programme the Hartlepool defeat was down to "two Cs: Covid and Corbyn".