The 24-year-old joined Spurs on a free transfer in July after taking advantage of a FIFA rule which allows foreign players in Russia and Ukraine to unilaterally suspend their contracts following the outbreak of war in the region.
Shakhtar are one of several clubs to have contested the changes to Annex 7 of the Regulations on the Status and Transfer of Players but those protests were dismissed by FIFA and also the Court of Arbitration for Sport.
Solomon was under contract with Shakhtar until the end of 2023, meaning Tottenham would have had to pay a transfer fee had Annex 7 not existed.
However, Palkin is optimistic Shakhtar will receive some money direct from Spurs regardless as conversations continue with the club's Director of Football Administration and Governance, Rebecca Caplehorn.
"We had conversations with Tottenham, with Rebecca," Palkin told ESPN. "We agreed she will send propositions.
"We are now in, let's say, a negotiation process with them. I hope and I believe we will find a solution. At this moment, it is quite early to tell something specific because we are on the way in these negotiations."
Talks took place around the friendly game played between the two clubs in August -- a match arranged to raise funds for Shakhtar's charitable foundation, Shakhtar Social, which supports the ongoing humanitarian crisis in Ukraine.
Sources have told ESPN that a share of any future transfer fee that Spurs receive for Solomon is one option, although Palkin is aiming to recoup the €6.5 million ($6.81m) Shakhtar paid Maccabi Petah Tikva for the winger in January 2019.
"We want to cover our losses," Palkin added. "For Tottenham, it is not big money. We paid €6.5m and I don't believe this is huge money for a club of Tottenham's level, for a club that participates in the English Premier League. I believe it is possible to cover these losses. I am not trying to generate some kind of profit.
"I believe we will find a solution. I don't know if they will accept €6.5m but I believe we will find a solution. From an image point of view, they don't need this kind of case.
"This kind of case smells not so good because when you sign a free guy who should go for €20m, it is not good. We played a good friendly game, they were open with us, they helped us etc etc. Therefore I feel these relations, we have good relations with Tottenham and I believe these relations will help us to find a good solution. I believe in a maximum of two weeks, we will find a solution."
They are suing Lyon, as Palkin claims they pocketed a €1m fee when the winger joined Leicester on a further loan in January, while there is another case against Tete and Galatasaray together after he joined the Turkish side in August using Annex 7. Tete had a contract with Shakhtar through to the end of 2023.
"When Tete left our club the first time under Annex 7, he joined Olympique Lyonnais and they then loaned him to Leicester for €1m.
"They abused Annex 7 because they cannot sell players because the player belonged to us. The player had a contract with us. It looks like they received unjust enrichment at the expense of our club.
"Now, we filed to FIFA to have restitution of this money. After Leicester, Tete moved to Galatasaray. We are filing against Tete but he has united responsibility with Galatasaray. In accordance with Annex 7, the player should send us an official letter before July 1 to say 'I would like to use Annex 7 to suspend my contract.'
"Tete didn't send this letter. We asked Tete to return back to the club many times and he ignored us. Because of his absence, we unilaterally terminated his contract. It means that actually Tete terminated this contract.
"He has a clause in that contract of €150m, you know. Now, Tete and Galatasaray altogether, they are in united responsibility for this kind of termination of contract."
Although FIFA have rejected any requests to amend or abolish Annex 7, Shakhtar are continuing to lobby for change through the European Club Association, whose president Nasser Al-Khelaifi is personally involved.
"I had discussions also with Nasser and he is helping us to try and have contact with FIFA and find some kind of solution, how they can help Ukrainian football," Palkin said.
"When we are fighting against Annex 7, we are fighting for and representing all Ukrainian clubs. We got seven letters from Ukrainian clubs, I sent them to Nasser. These letters contain the situations of these clubs because we are still waiting for help from FIFA.
"How is it possible from one side, you release all our players, contract finish and they become free for the market?
"Clubs sign our players for whom we paid millions of dollars or euros but nobody helps us. At the same time, FIFA push us to pay back all our debts.
"After the war started, I have up to €40m debts. This is debts for players who were released by FIFA and at the same time, I cannot sell players, I cannot loan players but at the same time I need to generate money for player. It is a ridiculous situation."
Sources at FIFA have told ESPN that each player's situation is judged on a case-by-case basis so players cannot automatically leave on a free transfer.
Various adjustments to Annex 7 have also been made with Ukrainian clubs' financial health in mind. They include players and staff wanting to use Annex 7 must inform their team in writing by July 1 and those who extended their contracts after March 7, 2022 cannot now suspend their agreements.