The new Fratton Park boss is desperate to reinforce his squad as he looks to pull off a miracle escape from the drop zone.
Chief executive Peter Storrie is confident Pompey will have their transfer ban lifted in time to make signings during January's window.
Striker Sturridge, 20, arrived at Stamford Bridge from Manchester City in the summer. But the fee is still to be decided by tribunal and he has yet to start a Premier League game for the Blues.
Even so, Chelsea may decide they need cover for Didier Drogba and Salomon Kalou when the pair are away on African Nations Cup duty with the Ivory Coast next month.
But that will not deter former Blues boss Grant. Portsmouth have scored just 13 goals in 15 league outings this season - and four of those came in one game against Wigan.
Grant is also prepared to use his contacts at the Bridge to lure Brazilian veteran Juliano Belletti to the South Coast.
The right-back, 33, has recovered from an Achilles injury suffered in August but is struggling to break into the Chelsea first team, despite the absence of first-choice Jose Bosingwa.
Belletti has started just two league games this term under Carlo Ancelotti and is out of contract at the end of the season.
Grant believes his versatility and experience could be an asset to a squad short of quality.
I was having a laugh while reading this, already the club has problems paying wages to existing players. Now you're expecting Chelsea players to take huge paycuts when they very likely have better chances elsewhere even if to get first-team football?