That is despite the 20-year-old currently being on loan with Russian side Lokomotiv Moscow.
The Daily Mail says Saints want to negotiate a surprise permanent deal for the English attacking midfielder.
Anjorin is currently back at Chelsea recovering from a minor injury and is expected to return to Russia at the end of February.
It is believed Lokomotiv would allow Anjorin to leave if a deal suited all parties.
Broja is on loan from the Blues, but it is believed Southampton are also pushing to sign him on a permanent deal.
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