Both Dumfries and Galloway Council (DGC) and National Museums Scotland (NMS) hope to secure the artefacts.
The Scottish government has been asked to intervene to resolve the situation.
However, a spokesman said the decision lay with the Queen's and Lord Treasurer's Remembrancer and urged both sides to continue discussions.
The council hopes to secure the hoard to go on display at a new gallery being built in Kirkcudbright.
A campaign has also been launched locally to ensure the treasure - discovered in Galloway - returns to the region.
For its part, NMS has said it believed it had put forward a "mutually-beneficial and positive proposal".
It would allow part of the hoard to go on display in Kirkcudbright and, on occasion, the whole collection to be hosted by the gallery.
It had been hoped a joint bid could be agreed, but so far that has proved impossible.
A Scottish government spokesman urged the two groups to hold further talks to try to resolve the situation.
"NMS has proposed a collaborative approach with DGC which guarantees the long-term display of a significant portion and, for specific periods of time, all of the Galloway hoard in Kirkcudbright Art Gallery," he said.
"The Scottish government would encourage continued discussions to ensure the best outcome for the long-term conservation of the hoard and public access to it across Scotland.
"The decision on allocation is made by the Queen's and Lord Treasurer's Remembrancer and the advice of the Scottish Archaeological Finds Allocation Panel, not by ministers."
The panel is due to discuss the hoard's future later this month.
The Scottish government has said talks should continue between organisations bidding to host a Viking treasure hoard to ensure the "best outcome".