Joshua, the IBF heavyweight champion, had agreed terms to face Klitschko on 10 December but the Ukrainian will not fight unless the WBA title vacated by Tyson Fury is on the line.
"I know I can knock out Joshua because I've done it before," said Price.
"I put him down [in sparring] and he had to get helped out of the ring."
Joshua's mandatory challenger is Joseph Parker, but the New Zealander will instead fight Mexican Andy Ruiz for the WBO title, which was also vacated by Fury.
With Parker out of the picture, Joshua, who has 17 knockouts in as many victories in the professional ranks, can pick a challenger for his IBF belt.
Liverpool's Price, 33, has never fought Joshua in a competitive contest but the two did meet in a sparring session while Joshua was on Team GB's Olympic boxing squad.
Price claims to have floored his London rival in 2011, a year before he won a gold medal at the London Olympics.
"I've kept it to myself for years because I felt bad for doing it," Price, who won a bronze medal at the 2008 Olympics, told BBC Sport.
"And I wanted to protect the lad from people shouting him out because he got knocked out in a gym with 18oz gloves.
"People think Anthony Joshua is this colossus of a man with a great physique but I am convinced it can go wrong for him in a fight with me."
Price has had to rebuild his career after two knockout defeats by American Tony Thompson in 2013 before being knocked out again by German Erkan Teper. Both opponents later failed drug tests.
"I understand why a lot of people will write me off," said Price, who has 18 knockouts in 21 victories.
"The only thing I've done to not deserve this shot is to lose to two drug cheats and I've worked hard to recover my career so I think I do deserve it."
Former British heavyweight champion David Price says he should be Anthony Joshua's next opponent if a fight with Wladimir Klitschko collapses.