City are looking at options to improve their forward line, with Savinho - and potentially Omar Marmoush - set to join Tottenham.
There have not been any club-to-club talks between City and Liverpool at this stage for Gakpo, who is under contract at Anfield until 2030.
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Liverpool do not want to sell Gakpo as things stand and would likely need to bring in two wide forwards to be comfortable with letting him go.
The 27-year-old started and scored in the Reds' Premier League opener against Newcastle last weekend.
They have been pursuing deals for PSG duo Bradley Barcola and Ibrahim Mbaye, while they have had multiple offers rejected for Brighton's Yankuba Minteh.
Man City have already verbally agreed a deal with Palmeiras to sign Allan for £34m as a replacement for Savinho. Spurs have agreed to pay up to £85m for Savinho, who had a medical last week.
Sky Sports News reported last week that a deal for Marmoush was very close to being agreed by City and Spurs. He was named on the bench for City's Premier League opener against Bournemouth but did not come on.
Man City reach Allan agreement
Latest from Sky Sports News' Ben Ransom and Zinny Boswell:
Manchester City have reached an agreement with Palmeiras over the signing of Allan.
The deal for the winger is worth up to £34m. City are paying fixed fee of £32m plus £2m in add-ons.
Both clubs have a verbal agreement, but are yet to exchange documents for the 22-year-old's transfer.
The player will be given permission to travel for a medical once this process has been completed to the satisfaction of both clubs.
Allan is being brought in to replace his fellow Brazilian Savinho, who is set to join Tottenham in a deal worth up to £85m.
