According to the Sun, Celtic manager Brendan Rodgers still hopes Boyata will sign a new contract at the club, with just one year to run on his current deal.
However, the 27-year-old's performances in Russia this summer have reportedly attracted attention from a host of clubs around Europe, with Lazio rumoured to be among the potential suitors for Boyata.
While Celtic are reluctant to sell Boyata, they could be left with no other option should he choose not to sign a new contract.
The defender put contract talks with the club on hold prior to leaving for the World Cup and could now be tempted by offers from elsewhere.
The Sun claim Fulham are prepared to offer Boyata a significant pay rise should he move to Craven Cottage this summer and the allure of Premier League football could well have an impact on the player's final decision.
Fulham are attempting to bolster their squad ahead of their return to the Premier League, having secured promotion via the playoffs after narrowly missing out on automatic promotion.
The centre back made his Belgium debut in 2010 but only returned to the national side after a lengthy absence in 2017, a direct result of the significant improvements he has made as a player while at Celtic.