The Hammers' player of the year is widely expected to move back to the Premier League this summer after the club were relegated, with Tottenham his likely destination.
But new manager Sam Allardyce will return from holiday next week and one of his first tasks will be to speak to Parker and find out what the future holds for the 30-year-old England international.
Down and out: West Ham are hopeful of keeping hold of Parker, despite their recent relegation
'Until Sam gets back and speaks to him we haven't given up on the possibility that we can keep Scott Parker,' co-chairman David Gold said.
'That will be an issue for Sam. He will be back in the office on Monday and I guess that's when the hard work starts.'
If Parker does move on, Allardyce has a ready-made replacement in midfield in the shape of Kevin Nolan.
The 28-year-old has dropped down a division to link up with his old Bolton boss after leaving Newcastle for a fee of around ?4million.
International duty: Parker's place in the England squad alongside the likes of Wayne Rooney could be under threat if he remains in the second tier of English football
And Gold feels his club have pulled off a major coup in landing a player who scored 12 top-flight goals last term.
'He is a quality, quality player,' added Gold. 'The relationship between him and Sam is very important. He's captaincy material, he's got that determination, will and desire to win which Sam has and the owners have.
'I know the fans are delighted, it's an absolute coup for West Ham.'If we were still in the Premier League you would expect West Ham to be pursuing a player of his quality so to be in the Championship and persuade a great player to come to us...this is the player we can build our squad around, an iconic player.'
Coup: West ham co-owner Gold is delighted with the signing of Nolan
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?Explore more:People: Kevin Nolan, Sam Allardyce, David Gold Places: Newcastle, United Kingdom
Source: Daily Mail
Source: Daily Mail