Messi's current deal at Camp Nou expires in 2018 and, with talks over an extension ongoing, it was reported on Friday that the five-time Ballon d'Or winner had turned down an offer worth between €30 million ($44.47 million) and €35 million annually ($51.88 million).
The 29-year-old shrugged off the speculation to score twice as Luis Enrique's men beat Villarreal 4-1 and Mestre took the opportunity after the match to refute suggestions that discussions with Barca's prized asset were on the rocks.
"The truth is everything looks very good," he told TV3.
"Everything will end up well. If they're saying there are problems [with the talks] in Madrid, then I'm delighted to be able to contradict them.
"Lionel Messi's contract renewal is on track. I'd say we've hit the final straight and in a few weeks there will be news. I cannot give an exact date, but it won't be long."
Fellow star forwards Neymar and Luis Suarez have both penned contract extensions this season and club president Josep Maria Bartomeu has confirmed Barcelona's intention to make Messi the best-paid player in the world when he commits to fresh terms.
Messi scored his 500th goal for Barca in the thrilling El Clasico win over title rivals Real Madrid at the end of last month and he is on 51 for the season after his exploits against Villarreal.
As Barcelona try to keep the heat on Real Madrid in the battle at the top, added motivation may come in looking to end Luis Enrique's reign with a third title in as many years.
The coach confirmed in March that he will depart Camp Nou at the end of the season, with the likes of Ernesto Valverde and Ronald Koeman linked to the job.
Luis Enrique's assistant Juan Carlos Unzue has also been talked up as a potential number one by Barca players, and Mestre says a decision has been made.
He said: "We have already chosen his replacement but we won't communicate our decision right now out of respect for Luis Enrique."