And Odom, sources said, is giving the offer real consideration after Minnesota's Kevin Love turned down the chance over the weekend to join Williams with Istanbul-based Besiktas.
Besiktas has been trying for days to acquire a top-flight big man to pair with Williams, who erupted for 50 points Tuesday in a EuroChallenge game. Yet in that victory over Gottingen of Germany, Besiktas lost Cleveland Cavaliers forward Semih Erden to a thumb injury that only increases its need for a frontcourt boost.
According to media reports Thursday in Turkey, Besiktas also is in advanced talks with Williams' New Jersey teammate Brook Lopez, because Lopez would be a more traditional center to replace Erden.
Turkish journalist Mete Aktas reported later Thursday that the club remains in talks with both Odom and Lopez, and that Besiktas coach Ergin Ataman prefers Odom and would sign him if Odom decides he's prepared to make the leap abroad.
Besiktas can sign only one more foreign player in addition to Williams. But this week's sudden resumption in talks between the league and locked-out players figures to delay any decision from either player.
Besiktas also has talked to Chicago Bulls teammates Luol Deng and Carlos Boozer about a contract that would include an out to return to the NBA as soon as the lockout ends, but the insurance costs in both cases appear to be prohibitive, given that both players have expensive long-term contracts. Boozer has four years and nearly $61 million left on his contract, while Deng is still owed just under $40 million in the next three seasons.
It's believed that Gortat -- owed just under $22 million in the next three seasons -- is highly interested in the job, which is why he has been widely described in Turkey as the most likely newcomer from the NBA to join Williams.
Odom's contract, though, would be the most manageable insurance-wise for Besiktas, with just two years left worth slightly more than $17 million, but only a partial guarantee in 2012-13.
Odom turned 32 earlier this month and is considering whether to accept the proposal, sources said. Odom is well-regarded in Turkish basketball circles following his strong play in Istanbul in summer 2010 for the victorious Team USA at the world championships.
NTV Spor reported Friday that Love and Deng were Besiktas' top two targets in its hunt for an elite forward to play alongside Williams.
"I didn't feel it was the right decision for me at this time," Love said Sunday in explaining his decision to pass.
Love, however, did say playing abroad, if the lockout drags on, "is still very much an option" he's considering.
Boozer is reportedly open to making the jump to reunite abroad with Williams, his former Utah Jazz teammate, if the insurance obstacles could be sorted out.
But questions remain about Besiktas' budget after two WNBA stars (Amber Harris and Jantel Lavender) recently left the club's women's team in a dispute over late payments.
Yet the fact remains that such doubts about finances did not prevent Besiktas from landing Williams amid considerable skepticism, or making well-publicized runs during the summer at Kobe Bryant and Kevin Durant. Besiktas also briefly employed Allen Iverson last season.
Besiktas' wide-ranging search for size to complement Williams, according to Aktas, also featured discussions with Memphis'
Zach Randolph, Dallas' Tyson Chandler, Chicago's Omer Asik, Chris Kaman of the Los Angeles Clippers and Andrew Bynum of the Los Angeles Lakers.
Senior writer Marc Stein covers the NBA for ESPN.com.