Chalov, 21, has been linked with a move to England throughout the Premier League summer transfer window, which closes at 5pm UK time on Thursday.
London club Palace lodged an initial offer of around £15 million for the forward in July, but were rebuffed by the Moscow club.
They are believed to have returned with an improved bid of a reported £23 million on Wednesday in a an attempt to lure CSKA into a sale, although it was again rejected.
The Moscow club are said to have wanted around £28 million for the forward, who netted 15 goals in 30 league games last season, and also scored in the shock 3-0 win against Real Madrid in the Champions League.
Chalov himself is said to have been keen on making the switch, although the CSKA management were reportedly concerned they would not find a suitable replacement for the young star.
Chalov came through the youth ranks at the club, and has also made two appearances for the Russian national team.
While not the biggest or most powerful forward, his technique and ability to find space are superb, as is his finishing.
Other clubs said to have been following the forward’s development include Premier League giants Manchester United.