Observers say he does not have enough support and elections will be called.
Reports suggest the election will be called for 20 September.
If Mr Meimarakis fails to form a government, the chance to do so will fall to the far-right Golden Dawn party, who finished third in January`s elections with 6.3% of the vote.
They, too, are unlikely to be able to gain enough allies to establish a government.
Both parties can waive the right to negotiate and allow the president to approve a snap election. But reports in Greek media said Mr Meimarakis would try and use his mandate to form a government in the next few days.
Mr Tsipras will lead his leftist Syriza party into the polls, but he has faced a rebellion by some members angry at the bailout`s austerity measures.
He had to agree to painful state sector cuts, including far-reaching pension reforms, in exchange for the bailout - and keeping Greece in the eurozone.
Greece received the first €13bn ($14.5bn) tranche of the bailout on Thursday after it was approved by relevant European parliaments.
It allowed Greece to repay a €3.2bn debt to the European Central Bank and avoid a messy default.
The overall bailout package is worth about €86bn over three years.
`I want to be honest`
Alexis Tsipras made the announcement in a televised state address on Thursday.
"The political mandate of the 25 January elections has exhausted its limits and now the Greek people have to have their say," he said.
"I want to be honest with you. We did not achieve the agreement we expected before the January elections."
Mr Tsipras said he would seek the Greek people`s approval to continue his government`s programme.
Greece will be run by a caretaker government ahead of the polls.
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