Wilson rejoined the Reds in Austria at the start of the week after his post-Euro 2020 break, but was omitted from the 27-man squad to play Mainz on Friday evening.
The 24-year-old is surplus to requirements and negotiations have been ongoing regarding a permanent move away from Anfield, with Benfica and Brentford among the clubs interested.
It is Fulham who are set to land the playmaker, however, with Liverpool close to agreeing a £10 million deal for their long-serving academy graduate.
The development comes despite previous claims that the Reds believe Wilson’s value had increased after another season on loan in the Championship with Cardiff, having turned down a £12 million bid from Burnley a year ago.
Wilson has made just two first-team appearances in his time with the Reds – who he joined at under-9 level – with those coming almost four years apart.