Mourinho began his managerial career with Benfica in 2000 but took charge of only 10 games and left after a dispute with the club president.
He then made his name with Porto between 2002 and 2004, winning six trophies – including the Champions League in 2003-04.
Since leaving his homeland in 2004, Mourinho has managed Chelsea, Inter Milan, Real Madrid, Manchester United, Tottenham Hotspur, Roma and Fenerbahce.
Should Benfica appoint Mourinho, he would face Chelsea in the Champions League at Stamford Bridge on 30 September.
Benfica also travel to Newcastle United and host Real Madrid later in the league phase of the competition.
Benfica are sixth in the Primeira Liga, five points adrift of leaders Porto, but have a game in hand.
They finished second in the top flight last season – two points behind Sporting – under Lage.
The move to part company with Lage coincides with Benfica preparing for their presidential election on 25 October.
Joao Noronha Lopes, favourite to win the election, has his sights set on Manchester United manager Ruben Amorim and was at Etihad Stadium on Sunday to watch Portuguese.
Costa said the decision to make a managerial change is not motivated by his attempt to get re-elected.
“I’m not here to save any positions,” Costa said.
“I haven’t even gone anywhere to campaign yet, because my goal, as president of Benfica, is to ensure the best future for Benfica, regardless of what happens in the elections.
“And this decision-making has to do solely and exclusively with Benfica’s sporting season, and to avoid jeopardising Benfica’s sporting season. I’ve never put my interests ahead of Benfica’s, and I’m not going to do so now.”