The club are close to appointing Luciano Spalletti for a second spell in the Italian capital, according to media reports.
Spalletti, who coached Roma between 2005 and 2009, wrote on his Facebook page late on Tuesday: "I'll finish what I've started!" and posted a photo of himself with Roma captain Francesco Totti.
After briefly leading Serie A early this season, Roma have slipped to fifth after one win in their last seven league games.
"AS Roma announces that it has relieved the first-team manager Rudi Garcia and his assistants Frederic Bompard and Claude Fichaux," the club said in a statement on its website (www.asroma.com).
Garcia had been in charge of Roma since 2013.
"We have had positive times together, but we think this is the right moment to change," president James Pallotta said.
Spalletti has been out of work since being sacked by Russia's Zenit St Petersburg in 2014.
Reports in Italy say he has agreed personal terms and accepted an 18-month contract. The appointment is expected to be announced on Thursday.
Roma's indifferent form has seen them slip seven points behind leaders Napoli in Serie A and they were eliminated from the Italian Cup last month by Serie B team Spezia.
They qualified for the last 16 of the Champions League despite only one win from six Group E games, finishing second behind Barcelona. Roma now face Real Madrid over two legs.
(reporting by Isla Binnie, additional reporting by Gavin Jones in Rome and Simon Jennings in Bengaluru, editing by Justin Palmer)
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