The 23-year-old, who joined the Swans for £200,000 from Wrexham last year, has recently been on loan at Fleetwood Town.
York boss Gary Mills moved for Obeng, who has yet to make his Premier League debut, after Lanre Oyebanjo was sidelined for around six weeks with a fractured skull, sustained in Saturday's 1-1 draw at Gillingham.
The Minstermen are also expected to complete a deal for Institute defender Dean Jarvis, 20, before the transfer window closes.
Former Northern Ireland U21 international Jarvis will join up with former Derry City striker David McDaid if he moves to Bootham Crescent.
McDaid, 22, has signed a two-year contract after York paid Derry an undisclosed compensation fee.