The Hammers axed Gianfranco Zola on Monday after 18 months in charge following a turbulent season which saw the east London club narrowly avoid relegation and the Italian coach repeatedly at odds with new owners, David Sullivan and David Gold.
Grant has been linked with the Upton Park hot-seat along with former Hammers favourite Slaven Bilic and ex-Manchester City boss Mark Hughes.
The Israeli guided Pompey to the FA Cup final against all the odds, with the club crippled by financial problems which led to administration, a points deduction and relegation from the Premier League.
Grant agreed to take over as manager following the dismissal of Paul Hart in November, stepping in from his role as director of football.
The 55-year-old is contracted to Fratton Park until 2012. However, it is unclear whether that deal remains valid, following the late payment of wages, and Grant insists he wants assurances of the stability of the club before he will commit to taking charge of the side in the Championship.
Andronikou, though, insists were West Ham to appoint Grant, Portsmouth would seek reimbursement.
"It is like any compensation claim - he has three years left so we would try to hang out for as much as possible," Andronikou said in the Portsmouth News.
West Ham are desperate to slash their wage bill this summer, with Sullivan declaring every player, other than captain Scott Parker, was up for sale "if it was the right bid".