Barca's 3-1 victory over Juventus in the Champions League final meant they became the first team to win their domestic league and Cup and Europe's elite club competition in the same season on two occasions, repeating their feat from 2009.
Luis Enrique is coming to the end of the first year of a two-year contract but was clearly unhappy with the sacking of sporting director Andoni Zubizarreta, the man who appointed him, in January.
He fended off several questions about his future at a news conference after Saturday's final, saying he was only focussed on celebrating the treble.
The uncertainty has been amplified by a looming presidential election and a new chief in place of Bartomeu might want to bring in his own man.
Bartomeu told Spanish radio on Monday he would offer Luis Enrique a contract extension if re-elected.
"There is no need to suffer, Luis Enrique will continue," Bartomeu said.
"His season has been 10 out of 10 and there is no reason to break the contract," he added.
Barca celebrated their magnificent season with an open top bus tour through the Catalan capital on Sunday, culminating in a spectacular party with fans at their giant Nou Camp arena.
They also announced the signing, pending a medical, of Sevilla midfielder Aleix Vidal, who has impressed for the Europa League winners this term and last month earned a first call-up to Vicente del Bosque's Spain squad.
(Writing by Iain Rogers, editing by Amlan Chakraborty)
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