Small cut ties with Everton earlier this week when he terminated his scholarship contract with the club two years early.
The 17-year-old was wanted at several clubs in the Premier League even before he decided to walk out of Everton this week.
Chelsea, Leeds and Aston Villa are amongst the clubs who have been keeping tabs on his situation and are interested in becoming his new home.
But Southampton believe they have their noses ahead in the race to sign Small in the final week of the window.
The Saints are certain they have managed to fight off competition from Chelsea, Leeds and Villa for the young full-back.
The south coast club are now looking to work out a compensation fee with Everton to get their hands on Small.
They want to agree a fee with the Merseyside giants without the matter going to a tribunal.
He became the youngest player to make his debut for Everton when he featured in the FA Cup against Sheffield Wednesday in January.
Southampton want Small to walk straight into their first-team squad to battle for a place in the team.
