As the Reds reported to their training camp on Saturday, one of the most notable absentees was Wilson, who was expected to link up with the squad after his loan with Bournemouth.
The 23-year-old has not travelled to Saalfelden, which prompted further speculation over his future, with a permanent move from Anfield possible.
Leeds and Newcastle have been touted as potential buyers, with Liverpool not open to loaning him out for another season, following spells with Hull, Derby and the Cherries in recent seasons.
But Wilson’s absence in Austria comes due to an injury, rather than any final decision over his future with the club, according to the Liverpool Echo.
Paul Gorst reports that both Wilson and Trent Alexander-Arnold did not travel with the rest of the squad as they undergo rehabilitation for minor issues, instead continuing to follow their individual training programmes at home.
Neither are considered doubts for the Community Shield clash with Arsenal on August 29, though it is likely neither starts, with Liverpool not due to return to Melwood until the start of next week.
There remains a strong chance that Wilson moves on if the price is right, but it could be that Klopp is still eager to take a closer look at the Welshman in training.
With the transfer window not due to close until October 5, there is no rush to conclude deals before the start of the season, though all parties would benefit from the clarity, and the club could reinvest any funds brought in.
Given the increased intensity of the schedule in 2020/21, it may be that Wilson is seen as a useful option for rotation, with Klopp describing the eight months ahead as “the most intense season probably of our lives.”
Liverpool are claimed to value the Wales international at around £20 million, though they may be forced to drop their price tag if they are hoping to sell.
Wilson did finish last season as Bournemouth‘s second-top goalscorer, with seven in 31 league appearances, but he was rarely trusted to play the full 90 minutes by Eddie Howe as the Cherries suffered relegation to the Championship.
With many sides likely to struggle in terms of finances with supporters still not permitted to attend games, the calibre of clubs interested in Wilson may not be able to afford him.
This may not, therefore, be the last Liverpool supporters have seen of the No. 59 yet, with there still being a chance he adds to his sole career outing for the club.