Liverpool have increased their offer for Real Madrid midfielder Asier Illarramendi to €22m (£15.7m), according to Spanish daily Marca.

Illarramendi, 24, joined Los Blancos for a fee of €35m from Real Sociedad in 2013. He was predicted to be a long-term replacement for Xabi Alonso, but has failed to make the impact expected at the Santiago Bernabeu.

Marca claims Liverpool have already had an initial offer of €15m rejected for the uncapped Spaniard, but suggest the Premier League side have now increased that bid by almost 50%.

Real are happy to part ways with the player, who started just 10 La Liga games during the 2014-15 season under Carlo Ancelotti. However, the Italian's recently appointed successor Rafa Benitez will have final say on the player's future, although Marca stresses Illarramendi "will become a Liverpool player" if the former Napoli manager sanctions the move.

Liverpool have been very active so far this summer as Brendan Rodgers and the club's transfer committee attempt to address the shortcomings that saw the club slump to a sixth place finish last season.

Manchester City midfielder James Milner, Burnley striker Danny Ings and Bolton Wanderers goalkeeper Adam Bogdan will all join the club on July 1 on free transfers, although the Reds will pay a compensation fee for Ings.

The club confirmed their fourth signing of the summer on Saturday 20 June, bringing in 18-year-old defender Joe Gomez from Charlton Athletic.

Liverpool also remain heavily linked with a move for Inter Milan playmaker Mateo Kovacic, with Italian newspaper Tuttosport reporting on 15 June the club have lodged a bid of €22.6m, with another €3m to follow in add-ons.

While Kovacic subsequently denied talk of an agreement with Liverpool in quotes attributed to the same publication, the Croatia international refused to completely rule out the possibility of leaving Italy.