The Saints were hoping to complete the signing of Anjorin before the transfer window slams shut on Monday evening.
Anjorin spent the first half of the season on loan at Russian club Lokomotiv Moscow, who had the option of triggering a purchase clause, which they did not take up.
The father and agent of the young midfielder were at Staplewood earlier in the week to discuss a potential move.
However, it is now expected that Anjorin will not be moving to Southampton this window on a permanent deal.
The Saints are expected to return to the table for the youngster in the summer and they will be looking to sign him permanently then.
Several Championship clubs were also interested in taking the starlet on loan, with the Blues open to that possibility, but the Saints look in prime position to sign him.
The Saints are looking to make a permanent move for another Chelsea youngster Armando Broja, though even that attempt is expected to be made in the summer.
