The firm said it had signed a Letter of Intent to enter into exclusive negotiations with Liberty House Group.
More than 1,700 people are employed in the division, which has factories in Rotherham and Stocksbridge.
Steel union Community said it welcomed news of negotiations following "months of unnecessary stress and concern".
More on this and other South Yorkshire stories
The union's general secretary Roy Rickhuss said: "This is a positive step for the UK steel industry; however there remain huge challenges which government must address."
The union said it would be seeking urgent talks with Liberty House Group and would be asking what their plans were for investment, protecting jobs and providing decent pensions for members in retirement.
Tata Steel's UK boss Bimlendra Jha said the announcement was "an important step forward".
"We now look forward to working with Liberty on the due diligence and other work streams so that the sale can be successfully concluded," he said.
The Speciality Steels unit makes high-end components for the automotive, aerospace and oil industries.
In April, Tata sold its long-products division, based in Scunthorpe, to Greybull Capital, a UK-based investment firm.
Tata Steel has announced its preferred bidder for the £100m sale of its Speciality Steels division based in South Yorkshire.