The newly-promoted English Premier League agreed a £2m (Dh11.6m) fee for the player.
The club will also have the option to sign the 23-year-old England striker for £17m at the end of the season - if they manage to avoid relegation.
But reports from England today indicate the Euro 2012 forward will reject the bid from Sam Allardyce to move to London in favour of remaining at Liverpool.
The deal would represent a halving in the value of the striker, who joined Liverpool from Newcastle United for £35m in January 2011, but has struggled to find form at Anfield.
Brendan Rodgers, the new Liverpool manager, had originally said he would not consider a loan deal for the player, who had also been attracting the interest of AC Milan and his previous club Newcastle.
CSKA Moscow have signed Sweden international midfielder Rasmus Elm for a reported EU6m fee.
The 24-year-old - once hailed by national coach Lars Lagerbak as the country's biggest talent since Zlatan Ibrahimovic - has signed a three-year deal with the Russian Premier League side after being linked with a host of European clubs.
Elm had spent three years at AZ Alkmaar, where despite an injury-blighted first season he racked up 18 goals in 83 appearances.
Martin O'Neill, the Sunderland manager, has moved to spend some of the transfer money received from Al Ain with a £10m bid for Fletcher.
Aston Villa are also understood to be interested in the 25-year-old Wolves forward.