The 24-year-old joined from Swansea this month in a four-year £15m deal.
However, due to having surgery in May, he is not yet back to full fitness, and will be missing at least three of Fulham's opening Premier League matches.
Fulham boss Slavisa Jokanovic told the Evening Standard: "He won't be ready. We will be waiting between two and three weeks for him to start working with us."
"I don't expect he can help us before September."
Mawson is set to miss the opening game against Crystal Palace on August 11, before games against Tottenham Hotspur and Burnley.
Jokanovic has looked to add reinforcements to his back line, with Arsenal's Calum Chambers being linked with a loan move to club.
This is the first time Fulham have been back in the Premier League since the 2013/14 season, after they beat Aston Villa in the play-off final.