Dutch forward Bas Dost and three other players informed Sporting on Monday that they were terminating their contracts, joining winger Daniel Podence and Portugal goalkeeper Rui Patricio, who quit on June 1.
Dost was injured during an invasion by hooded supporters, some armed with sticks, at the club's training camp on May 15, during which the dressing-room was vandalised.
Some players accused the president of "psychological violence" against them and inciting anger among fans.
De Carvalho had rejected calls by senior club members to stand down, accusing his critics of blackmail and "a concerted attack on Sporting".
But after Monday's move by Dost, Gelson Martins, Bruno Fernandes and William Carvalho, the president told reporters: "We will resign if the athletes write a letter to Sporting saying, firstly, that if this management quits they will reverse their terminations and play for Sporting ... Secondly, if we run and get elected again, they maintain their contracts."
De Carvalho has refused to accept the legitimacy of a proposed vote on June 23 to depose him and his colleagues, following a tumultuous season which culminated in a shock defeat in the Portuguese Cup final by underdogs Desportivo Aves, five days after the fans' attack.
In April, the president threatened to suspend almost the entire first-team squad and lashed out at the players, calling them "spoilt children", according to text messages cited by Patricio in his resignation letter.
It is unclear whether the president's new offer will persuade the players to change their minds - none has been available for comment.
Sporting are one of Portugal's big three clubs, along with Benfica and Porto, having won the Primeira Liga 18 times, as well as the European Cup Winners' Cup, and have never been relegated in the league's 84-year history.
(Reporting by Andrei Khalip,; Editing by Neville Dalton)
