The Saints have been long-term admirers of both Celtic goalkeeper Fraser Forster and centre-back Virgil van Dijk, who only joined the club from Groningen last summer.
A bid in the region of £4.5m for Forster was tabled in the last week, but Southampton saw their approach knocked back and described as 'derisory' by the SPL champions.
Celtic were understood to be negotiating for a desired price of £8m, although the clubs will likely make an agreement in due course, with Southampton hopeful that £6m will persuade the Scottish club to allow the England international to depart Parkhead.
It is understood, however, that Celtic are reluctant to sanction any departures until they have secured the prize money for Champions League qualification, with their return tie against Legia Warsaw scheduled for August the 6th.
Southampton, on the other hand, are keen to have van Dijk at the club before that date, despite an acceptance that they may have to wait on Forster's signature.
The presence of Artur Boruc means that the Saints are in less of a rush to complete the deal for Forster, whilst the departure of Dejan Lovren makes the addition of a new central defender a priority for Ronald Koeman.
Koeman, who was born in Groningen, is an admirer of the player and in the build-up to the 2014 World Cup, suggested that the Celtic defender should make Louis van Gaal's squad to go to Brazil and represent the Netherlands.
Celtic's asking price for the twenty-two year old is in the region of £10m, although Southampton are hopeful that a compromise can be reached in order to swiftly complete the deal to bring the Dutchman to St. Mary's.
Would Celtic duo Forster and van Dijk improve Southampton?