The former England international has impressed at Coventry City, whom he guided to promotion to the Premier League last season, and is understood to have admirers among Palace's board.
According to The Times, Lampard and former Ipswich Town manager Kieran McKenna are two leading options should Palace decide to make another change in the dugout.
Sage arrived in the summer to replace Oliver Glasner, who left after leading Palace to Conference League glory. However, the Frenchman has endured a difficult start to life in England and is already under pressure to improve results.
Crystal Palace have conceded 17 goals in their first seven Premier League matches, including a 5-0 defeat against Brighton. They have also suffered a 3-2 loss to Lyon in the Europa League, increasing scrutiny of Sage's defensive approach.
The club have reportedly given the manager more time, but want to see an improvement at the back. Injuries and suspensions have disrupted his defensive selections, while attempts to recruit quicker central defenders during the summer did not produce the intended results.
Lampard's name is not new to Palace. The club considered him during the summer but did not meet Coventry's valuation, and he subsequently signed a contract extension running until 2029.
His work at Coventry has strengthened his reputation after spells in charge of Chelsea and Everton. Taking Coventry into the top flight also gives him recent experience of building a team capable of competing for promotion, although a move to Palace would represent a different challenge.
McKenna is another candidate under consideration. He left Ipswich after their return to the Premier League, and Palace would reportedly need to account for compensation connected to his previous contract.
Palace have also explored other managerial options in recent months, including Andoni Iraola, who turned down the opportunity, and Dino Toppmöller.
For now, Sage remains in charge, but the pressure is mounting. With a demanding run of fixtures approaching, his ability to tighten the defence could determine whether Palace stick with their summer appointment or turn to one of the alternatives being discussed.