Pulisic lost his place to Sancho at Borussia Dortmund before his move to Stamford Bridge.
The England international is being linked with the Blues and Manchester United, but Burley says it shouldn't concern Pulisic.
"There are other areas of the pitch they might look to strengthen - midfield, striker, maybe another centre-half.
"They've got enough playmakers in my opinion. [Pulisic has] got a job on his hands like everybody else.
"He faced the same thing at Dortmund before he left because he had Sancho and [Jacob] Bruun Larsen, who'd ousted him from the team.
"It's not new for Christian Pulisic to be fighting for your place, that doesn't mean you're a bad player overnight.
"Yes it's a concern when you're not playing but I do think he showed enough and showed enough to Frank Lampard in the period that he was there that he's going to be a big part of the Chelsea plans going forward."