Roy Hodgson hopes to recruit a new goalkeeper, central midfielder and goalscorer in January, with the latter a particular priority as Christian Benteke - who has not scored since May - is their only senior, recognised striker.
An injury to the Belgian earlier this season led to Hodgson starting wingers Wilfried Zaha and Andros Townsend in an unorthodox front two, and while the 27-year-old is again fit, he is struggling to score.
Press Association Sports understands an approach has already been made for the £10million-rated Carrillo, 26, who has impressed for Ligue 1 champions Monaco following Kylian Mbappe's departure to Paris St Germain.
Following a challenging start to the season the Argentinian has scored five goals in Monaco's past nine games, and is considered the solution Palace require at a time when a further winger, Bakary Sako, is also being used as a striker.
Hodgson has remained supportive of Benteke, while also warning him Palace "cannot afford to carry anyone", and discussing his hopes for January, said: "We have only got one recognised centre-forward in Chrstian Benteke. Bakary Sako was signed as a winger.
"We have only got two goalkeepers in the club, which I don't think is anywhere near enough for Premier League football. We saw what happened when Wayne Hennessey had a back spasm the other day at West Brom.
"In central midfield we don't have enough cover, and with Luka Milivojevic out we find ourselves shorter there.
"Those are the positions we would like to strengthen and cover."
Palace visit Leicester in the Premier League on Saturday, when Benteke is again expected to retain his place despite Bako scoring as a substitute in Tuesday's 2-1 defeat of Watford.