The Brazilian is reportedly high on the Catalans' list of priorities in the summer transfer window however a place in Europe's premier club competition is expected to see them fend off potential suitors for at least another summer.
Victory over already relegated Middlesbrough on the final day of the Premier League season ensured the Reds grabbed the final place in the illustrious competition.
Coutinho, who scored a second half free-kick during the 3-0 win at Anfield, meanwhile enjoyed his most fruitful campaign for the club with 14 goals and nine assists in all competitions to further swell his importance to the Merseysiders.
Manager Jurgen Klopp has no intention of allowing Coutinho to leave Liverpool but that has not deterred speculation regarding the 24-year-old's future.
Spanish publication Sport understand Barca have identified Coutinho as the preferred candidate to replace Andres Iniesta or Ivan Rakitic, both of whom have struggled during a campaign where Luis Enrique's side failed to retain their La Liga title.
Coutinho is understood to have already agreed to move to the Nou Camp but Liverpool's €90m (£78.6m, $100.1m) valuation of the player means a deal is still some way from reaching a conclusion.
But now that the club have a spot in the Champions League for just the second time in eight seasons Rush believes that should be enough to warn off interest from afar.
"He's a quality player and he can play for any team in the world with the way he played this year, but if you're in the Champions League then I expect him to stay," Rush, who netted 346 goals across two spells at the club, told AS.
"It's another experience for him. There's a time and place for everything and Coutinho wants to play in the Champions League because he's a champion - but Liverpool will give him the chance to do that.
"I am pretty certain he will stay even though there are rumours about everything. The fans want him to stay. That experience of the Champions League with Liverpool will make him an even better player."