Allen, who joined from Swansea in 2012, wants first-team games after starting only eight in the league in 2015-16.
Liverpool rejected an £8m offer from Swansea City for their former player in June and it is understood the Welsh club are still interested in Allen.
Allen is open to the idea of staying at Liverpool should he be guaranteed a more regular starts at Anfield but Klopp preferred Jordan Henderson, Lucas, Emre Can and James Milner in midfield last season.
The Wales international has had a rollercoaster Liverpool career since joining the club for £15m in the summer of 2012.
He was part of the squad that helped Liverpool gain second place in the Premier League in 2013-14 but has often been the brunt of fan criticism.
Allen did score a late equaliser in a 3-3 draw with leaders Arsenal in January 2016 before scoring the winning penalty in a shootout against Stoke City to send Liverpool to a League Cup Final against Manchester City.
Allen is currently away on holiday and is not due back in Liverpool training until late July.