Watson's deal expires at the end of the season, but he says he has a "sense of unfinished business" at the club.
"I want to stay," the 25-year old told BBC Scotland ahead of Edinburgh's derby against Glasgow Warriors on Monday.
"At this point in time, I'm doing well for Edinburgh, and the best place for me to keep getting capped is playing in Scotland."
Edinburgh have finished no higher than eighth in the past five Pro12 seasons, and Watson, who played every minute of Scotland's three autumn internationals, wants to follow back-row colleague Magnus Bradbury in pledging his future to the capital side.
"Even though this is my sixth season, it's quite frustrating, because we win really tough games, and once we put together a season, this squad can achieve big things, and I don't want to miss that," he said.
"Definitely, there is a sense of unfinished business. I believe in the next two years we can achieve a lot if we hold on to all of our players."
Scotland flanker Hamish Watson says he "wants to stay" at Edinburgh beyond the end of his contract.