Reds boss Brendan Rodgers has faced potential competition from Arsenal who are also involved in a summer of rebuilding, but Gunners’ boss Arsene Wenger won't go over £6.5m for the keeper.
Sunderland rejected Liverpool’s £6.5m offer out of hand for a player who was pivotal in keeping the club in the Premier League last season.
But the Reds have not been dissuaded and have tabled several improved offers, rising to £8m early this week.
Yesterday, the Merseysiders were prepared to go to £9m, but Sunderland are holding out for a fee in excess of £10m for a player Steve Bruce paid £2m for, three years ago.