The West Ham midfielder has attracted bids worth £90m from Arsenal and Man City but Redknapp can ‘see a future’ for Rice in east London.
Arsenal have been confident of bringing Rice to the Emirates Stadium for several months, with Mikel Arteta a known admirer of the 24-year-old.
The Gunners have already submitted two bids for Rice but West Ham, looking for £100m, quickly rejected the offers.
This has given Manchester City hope of securing Rice’s signature and the Premier League champions made their first formal bid for the England star on Monday night.
While that also looks set to be knocked back by West Ham, Man City could well return with improved turns and possibly even blow last season’s title rivals Arsenal out of the water.
Redknapp admits Rice may end up leaving West Ham this summer but has not ruled out the chances of the midfielder staying put for another season.
‘West Ham are a big club with up to 65,000 fans in the stadium. I can see a future for him with the club,’ he told Claret and Hugh.
‘If the board were to offer him up to £250k a week and brought in three or four top players that may tempt him to stay where he is despite him wanting Champions League football.
‘The Hammers are in Europe and he won his first medal with us last season. I would really love to see the club push to keep him.
‘In reality it looks as though he will go to Arsenal as things stand and they do begin to look a very good team to whom he would be a big help.
‘I certainly see them as being the main challengers to Man City next season and beyond. They are a young and exciting team but I would have thought he would want to speak with Man City.
‘However, at the end of it all I want the Hammers to have a right go at keeping him. We are a big club and he could do far worse than to stay where he is.’
MORE : Man City make formal bid for West Ham star and Arsenal target Declan Rice
MORE : Arsenal defender Jakub Kiwior reacts to Jurrien Timber transfer links
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