Local newspaper reports in Berkshire on Friday suggest that Reading are unlikely to increase their £2m offer for the full-back, rejected last week by Nottingham Forest.
The other interest in the Welsh iinternational, together with a bid, came from Swansea City but the Liberty Stadium club now found themselves without a manager after Brendan Rodgers became the new boss of Liverpool.
His interest in Gunter may not follow Rodgers to Merseyside as he has inherited a squad which contains both Glen Johnson and Martin Kelly, both primarily right-backs and both part of the England set-up at Euro 2012.