Mike Hedges said party members should be offered a left-wing option at the leadership election, when it happens.
Stopping short of saying he would run, he said he was one of four AMs who had consistently backed Jeremy Corbyn.
Mr Jones has been Welsh Labour leader and first minister since 2009, and has not set a date for his departure.
He defeated rivals Edwina Hart and Huw Lewis for the Welsh Labour leadership in 2009 and said in March that at 50 years old he was younger than Theresa May and had given no thought to retiring.
On Thursday, Swansea East AM Mr Hedges also called for rule changes that could open the door to more candidates and potentially give a left-winger a clearer run for the leadership.
"When we have an election, when Carwyn decides to step down, we should have a left candidate like we did with Huw Lewis last time," Mr Hedges said.
"Traditionally leaders have gone after about 10 years, but it's entirely in Carwyn's hands," he added.
Welsh Labour is consulting on whether to introduce one-member-one-vote elections (Omov) for its next leader, as happens when the party elects a UK leader.
That would mean all the votes cast count equally, instead of the existing electoral college system which splits the party into three wings.
Mr Hedges is in favour of Omov and says candidates should be able to nominate themselves. Current rules require 20% of Labour AMs to support a candidate before they get on the ballot paper.
A Corbynite candidate should run for the Welsh Labour leadership when Carwyn Jones stands down, a Labour AM has said.