Mane was further back but the Senegal international suggested more success with European champions Liverpool could help him compete for France Football's prestigious prize.
The Reds, now a force at home and abroad, had four players in the top 10 and six in the top 20.
"That means we are a great team with great players and great quality," Mane told his club's website.
"The boys still have a lot to show and hopefully this season we can show it.
"I am very proud and happy to be named fourth in the world. I am looking forward and why not number one, maybe?
"I will do everything possible to win the Champions League and Premier League with Liverpool and we'll see what's going to happen."
Messi himself said it was a "shame" Mane had not finished higher in the aftermath of a 22-goal Premier League campaign.
The 27-year-old bounced back by starring in the 5-2 trouncing of Everton in the Merseyside derby and could start again when the leaders visit Bournemouth on Saturday.
"I think tiredness is in the head. I'm ready to go - I'm feeling fresh again," he said.
"I expect [to be rested at some point], but when they put me on the bench I respect it and I will be here to push my team-mates because we are Liverpool.
"Wednesday was a good example with Bobby [Firmino] and Mo [Salah], it meant that other players can get the job. I'm just here and whatever they decide, I am ready to go."