Daily Vecernje Novosti said on its website Partizan's board would meet by Friday to name Grant as a replacement for Aleksandar Stanojevic, who fell out with club president Dragan Djuric last month.
Grant steered Chelsea to the 2008 Champions League final and was in contention for the Premier League title to the last day of the season after taking over from Jose Mourinho at Stamford Bridge in 2007.
Grant's contract was terminated at the end of the season after he conceded both titles to Manchester United and after an eight-month stint at Portsmouth, he failed to save West Ham United from relegation last season.
Stanojevic won the Serbian league and cup double last season and also steered Partizan to the Champions League group stage for the first time since 2003.
Partizan headed into this season's winter break 10 points clear at the top but Stanojevic fell out with Djuric along with former managing director Mladen Krstajic, who was sacked on Boxing Day after a long-term conflict with the president.
While on vacation, Stanojevic said he would resign as soon as he returned to Belgrade but then made a U-turn, saying he wanted to stay until the end of the season to retain the domestic double.
The Serbian first division resumes on March 3.
