"Vera has been injured a lot this year," Tarpishchev said after leaving the world number 10 out of Russia's squad for next week's Fed Cup semi-final against Serbia.
"If she doesn't show her best at the French Open (in May), she won't be able to make our Olympic team. She must have a good claycourt season in order to stay in the top 10 and qualify for London."
Zvonareva, 27, won a bronze medal behind Elena Dementieva (gold) and Dinara Safina (silver) as the Russians swept the podium in the women's singles at the 2008 Beijing Olympics.
However, Zvonareva - Russia's second highest-ranked player behind world number two Maria Sharapova - has struggled for form this season after reaching two grand slam finals (Wimbledon and U.S. Open) in 2010 and being ranked number two in the world.
After missing Russia's 3-2 win over Spain in the Fed Cup first round in February, Zvonareva said she would be available for the semi-finals.
But Tarpishchev has decided to call up world number 21 Maria Kirilenko, 22nd-ranked Anastasia Pavlyuchenkova, Svetlana Kuznetsova (27) and doubles specialist Elena Vesnina for the April 21-22 tie at Moscow's Megasport indoor arena.
The Russians are bidding for their fifth Fed Cup title since 2004 after losing last year's final to the Czech Republic in Moscow. The Czechs are facing Italy in the other semi-final.