The chances of the 41-year-old being appointed the full-time boss in the summer are getting stronger following the Blues' progression into the Champions League final after the side's two-leg victory over Barcelona.
The former Premier League champions are also due to face Liverpool in the FA Cup final at Wembley on May 5.
"[Abramovich's] dream is the Champions League," Grant told BBC Radio Five Live. "They want recognition as a team in Europe. I hope he will win it.
"Roberto Di Matteo deserves one more year at least but Abramovich sometimes thinks differently. In my time, we also got to the final and he decided I will not continue."
The likes of Laurent Blanc and Jose Mourinho have been reported as possible targets for the Russian billionaire.