Liverpool managing director Ian Ayre says the club are delighted to have the Uruguayan back in action, and expects talks to begin over a new deal at the end of the season.
Suarez returned from his 10-match ban, received for biting Chelsea's Branislav Ivanovic, on Wednesday, when Liverpool faced Manchester United at Old Trafford in a third round Capital One Cup tie.
The Reds were beaten 1-0 by United after Javier Hernandez's winning effort. Suarez has also previously served an eight-match ban for allegedly racially abusing United's captain, Patrice Evra.
Ayre told Sky Sports News: "For all his controversy on the pitch, off the pitch he's a great guy, he's a family man and he's very well liked.
"He keeps himself to himself with his family, he's been committed he's worked hard, trained hard.
"It's been difficult, and been difficult for him really. It's always difficult being a footballer, and being a footballer like Luis Suarez you want to be on a football pitch.
"We sort of begin those type of discussions two years in from the end, so it'll be the end of this season. We'll do the same with everyone and at the right time we'll have the discussion with Luis and his advisors."
"I think it's been a challenge for him but he's got through it and I think, and he's demonstrated last night, for a first game back, for somebody who hasn't had the opportunity to play in the Premier League this season, he acquitted himself very well."
Suarez signed a long-term contract with the Anfield club in 2012, but expressed his desire to leave this summer, after failing to gain Champions League football with Liverpool. He was linked with a move to Real Madrid, and was subject to two rejected bids from rivals Arsenal, but ultimately he stayed.
Ayre admitted Suarez's latest actions hurt the global image of the club, but said Suarez is looking to draw a line under it.
"Of course, any types of incident of that nature are damaging to a brand, but Liverpool Football Club is way, way bigger and always will be way bigger than any one player or one individual," Ayre added.
"Luis is a street fighter and we had to deal with it. We've worked a lot with him since the Ivanovic incident and he's back to what he's best at - terrorising defenders.
"We all move on together. Hard work and lessons have been learnt post-Evra and we now have a process in place for any crisis. Not a Suarez crisis, just anyone."
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