The 26-year-old Togolese international will travel to Madrid on Wednesday to undergo a medical.
Real will have the option to sign Adebayor on a permanent basis at the end of the season and Spanish media reports said they could pay around 15 million euros for him.
The powerful front-man will provide Real coach Jose Mourinho with an extra attacking option following a back injury to Argentine striker Gonzalo Higuain that will sideline him until around April.
Mourinho has been pushing the club for another centre-forward since arriving from Inter Milan last year and with Higuain's injury and Karim Benzema's failure to live up to expectations the pressure had grown during January.
France striker Benzema has only scored two league goals this season as Real chase La Liga leaders Barcelona, prepare for a King's Cup semi-final and the Champions League knockout stages.
Real's first choice as a stand-in appeared to be their former striker Ruud van Nistelrooy but SV Hamburg refused to release him.
Moving for Adebayor, the 2008 African Player of the Year, had the added bonus that he is a free to play for Real in the Champions League.
Adebayor started his career in France with Metz and AS Monaco before making his name after signing for Arsenal in 2006.
He scored 62 goals in 142 appearances in all competitions with Arsene Wenger's side before moving to Manchester City on a five-year contract in 2009 for around 25 million pounds.
He managed 19 goals in 36 starts for City but has dropped out of the reckoning this season and was pushed further towards the door when City signed Bosnian striker Edin Dzeko this month.
