"Unfortunately we estimate he will be out for somewhere between three and four months," Lampard told reporters Friday, according to the club's website. "I am obviously very disappointed as we all are here, and Billy is himself. I spoke with him yesterday at length and just gave him support, being there for him."
The 19-year-old was injured during a collision with Palace's Luka Milivojevic in the 19th minute of Tuesday's 3-2 victory at Selhurst Park.
The injury cuts short a debut senior season that saw the Glaswegian shine in March, with successive Man of the Match showings in a cup win over Liverpool, and a comprehensive 4-0 battering of Everton, his first league start.
In 11 outings in all competitions, Gilmour was a competent deputy in the center of a three-man midfield in Jorginho's absence. With Jorginho's place at Stamford Bridge uncertain amid speculation about a potential return to Italy, Gilmour was set to take on increased responsibilities next season.
"Part of our job as players is to get on with things, he will learn that very quickly, and his attitude will get him back as quick as he possibly can," Lampard added.
Chelsea travel to Bramall Lane to face Sheffield United on Saturday, looking to maintain their third-place standing in the table with an eye on securing a Champions League spot for next season.