"Avram is a very experienced and respected manager who has managed at the highest level. The board believes he is the man to help steer the club out of the relegation zone," chief executive Peter Storrie said on the club's website.
Grant will take charge of the league's bottom team for the home game against Manchester United on Saturday in his role as director of football. The club will then apply to switch his work permit status to manager.
"He knows the club, the players and the set-up at Fratton Park, so it was the logical move to make him the next manager, once the board had decided to relieve Paul Hart of the role," Storrie added.
The 54-year-old former Portsmouth technical director returned in October shortly after Saudi Arabian businessman Ali al Faraj's takeover of the south-coast club.
Al Faraj bought 90 percent of the shares last month from Sulaiman al Fahim, who had taken over in August but struggled to refinance Portsmouth's debts.
"We will have our first official board meeting on December 1 and I am giving back the property rights to the club so they can upgrade the current stadium," Fahim said on November 15.
In September, payments to some of the first team squad were delayed, highlighting the club's problems.
