West Brom have sacked Nicolas Anelka citing "gross misconduct" after the striker claimed on Twitter he had quit the club.
Mauricio Pochettino expects to be at Southampton next season and is looking forward to working with chairman Ralph Krueger.
The former England Under-19 international will spend an initial month with the Minstermen.
The 23-year-old right-back, yet to make his senior debut for Swansea, was signed from Welsh rivals Wrexham last year for £200,000.
Obeng becomes York's fifth recent signing following the arrivals of striker David McDaid from League Of Ireland Premier Division side Derry, teenager Jack O'Connell, on loan from Blackburn, fellow defender Jameel Ible, 19, from Salford City and striker Ben Everson from Icelandic side Breidablik.