The midfielder has spent 13 years with the Swans, but struggled for appearances in the season just gone.
Britton made just nine appearances for Swansea this season, his lowest ever at the Liberty Stadium, and Monk has admitted to the Swans favourite that he may struggle to get game time next season.
"I said to the manager that I can't do another season not playing," Britton says in the Post.
"He said that he sees things being very much the same next season, if not (worse) with me being out of the squad completely because he is trying to push young lads like Jay (Fulton) and Matt (Grimes) through.
"They are the future of the club, I understand that.
"Jack Cork has come into the club and he has done brilliantly and the team have done brilliantly, so I can't complain."
The 32-year-old has over 400 appearances for the South Wales club, and is one of a handful of players to have famously played in every division for the Swans.
Britton has a contract until 2017, but the midfielder's phenomenal record, stretching over 13 seasons at the club, suggest he may released on a free if he wished to try something new.
"My first choice will always to be play for Swansea," the Englishman added. "I would love to finish my career at the club.
"But I still feel I have got something to give and I am not going to be happy not playing.
"I want to play here, but if the chances of that happening are limited I will have to look at what options are out there and I have spoken to the manager about that."
As he is about to turn 33 in September, it is understandable that Britton is determined to make the most of his last few seasons as a footballer, and though he admitted in the interview that links to the MLS hold no substance right now, a switch to a league that is less frantic may be something that could prolong his career.