The versatile Australian international nominally plays as a centre-half, but is also able to take a right-back or holding midfield role. The 25-year-old was reportedly close to agreeing a deal with Swansea last week, but Celtic have been made aware of the situation because no agreement between the clubs has taken place.
Middlesbrough are understood to be willing to part with Williams because of his wages, which are around £20,000 a week. Celtic may find difficulty in matching those terms, but the promise of Champions League football and trophies has often proven to be a substantial bargaining chip for Celtic.
Celtic are looking to provide cover and competition for regular centre-half pairing Virgil van Dijk and Efe Ambrose, and highly-rated Williams is expected be a key cog in Celtic’s defensive machine should they succeed in prising him from Middlesbrough.
Meanwhile, Celtic have also been linked with Serbian midfielder Filip Djuricic. The Benfica midfielder, who moved to Portugal last summer from Eredivisie side Heerenveen, is reportedly unsettled and would be open to a move.
Djuricic may be best known to Scots as the man who scored twice against Scotland in a World Cup qualifier in Novi Sad last March.
The Serbian, who turns 22 next week, would come with a relatively high price tag - around £6 million.
Reports of interest for Steaua Bucharest midfielder Gabriel Iancu were widely reported in Romania, but rumours of a £3 million bid appear to have been tempered by Celtic’s interest in Djuricic.