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According to a TuttoMercato report, the midfielder was disappointed that no deal was agreed between Celtic and Torino for a summer move.
Martin O'Neill had confirmed there was interest from Celtic in the 22-year-old Lithuanian, but suggested Torino were adamant over a certain fee for the player.
In the summer, it had been reported Torino were holding out for £12m for the player. But no fee was agreed, with the player remaining in Italy.
It has now been claimed Celtic could return to the table for the player in the January transfer window, with Torino willing to negotiate at around the €8-10m price point, which would be around £6.75m to £8.5m.
However, it's unclear whether the player would still be keen on a move after his disappointment when no transfer could be sealed in the summer.
Discussing Celtic interest previously, Gineitis said: “I’m happy at Torino, I’m focused on my team.
“But whether you like it or not, you keep thinking about it in the transfer market. You know something could happen, there could be some movement.
“Playing for Celtic would be an interesting, wonderful experience. Also because testing yourself at a club like that would be truly stimulating.
“But for now, as I said, I’m at Torino and I’m thinking about Torino."
O'Neill had said: “We had more than an interest in him at the time.
“But it’s all very well having an interest and people feeling that they might want to come, you still have to negotiate with the club. That was a major thing at the end of the day.
“When clubs become adamant about a certain price for a player – and we are not in the market for overpriced players like in England – you have to deal with it as best as you can.”