The 20-year-old, who joined the Blues in 2012 from Anderlecht, is in Antonio Conte’s first-team squad this season.
Musonda is under contract until 2019 and according to the Evening Standard, Chelsea want to secure the winger’s long-term future.
They reckon Manchester United, Arsenal and Celtic have been tracking the Belgium Under-21 international in recent months.
The Standard say Chelsea held initial talks with Musonda at the end of last season but negotiations were pushed back.
Musonda reportedly wanted to see how much game time he would get under Conte this season.
The Standard add that Conte has told the winger he trusts him and he’s in his future plans.
Musonda revealed earlier this month that his debut goal for Chelsea in the Carabao Cup almost brought him to tears.
"It was an emotional moment, I can’t even remember it," Musonda said.
"I was very happy and you can see that.
"I was in tears to be honest.
"If the lads didn’t come and get me I don’t know what would have happened.
"The fans were great today. They gave me a big lift tonight and they were amazing."
