Phillipe Coutinho's transfer request was a surprise given it hasn't even been a year since he signed a five-year deal with Liverpool. While it is not ideal, keeping Coutinho around would only cause a distraction the team can ill-afford.
If Liverpool decides to sell Coutinho, the team could get a good return on its investment. The fee the organization could command for the talented Brazilian would be large enough for them to sign other players, like Paris Saint-Geramine winger Julian Draxler or Southampton center back Virgil van Dijk. Coutinho would be missed, but selling him could make Liverpool better overall.
Liverpool has had a rough time in the Premier League lately. The organization can't seem to keep their stars happy long enough for the team to legitimately compete for a league championship. Selling Coutinho would only serve the narrative that the team can't break into the elite class of European teams, which greatly affects Liverpool's ability to draw big names to the club.
Liverpool has to convince Coutinho he can win titles with the club, and be the centerpiece of the team. If Liverpool sells him, it's another star lost for the Reds, assuring their 27-year Premier League title drought continues.