The England international has been in fine form at the St Mary's Stadium this season and has been linked with the likes of Arsenal, Chelsea and Manchester United.
Last week manager Ronald Koeman admitted it would be difficult to retain the full-back and said Southampton face a battle to hold on to one of their prized assets.
Clyne insists that his focus remains on Southampton, but he stated his desire to win trophies and play in Europe's premier event.
"It's always good to have teams looking out for you, it shows how well you've been doing all season," he told Sky Sports.
"I am going to take that, and keep on performing and be the best player I can be.
"Right now I'm concentrating on playing for Southampton, there are still games to play, and try to push for that fourth spot.
"I am an ambitious person and I want to play Champions League football. To win the title and the FA Cup, that is what I want."
