The Ukrainian champions have seen Alex Teixeira, Fred, Dentinho, Douglas Costa, Facundo Ferreyra and Ismaily fail to return to the club due to political tensions in the country.
The club is yet to confirm the return of any of the players, who are thought to have stayed in France after Sunday's friendly with Lyon.
With the players still absent and problems in the region ongoing, Palkin reassured players the club would only play their upcoming domestic campaign in a city classified as safe by UEFA.
"I would like to stress straight away that the issue of players' safety is the main priority of our club," he told Shakhtar's official website.
"Shakhtar will base itself, train and play only in those cities where the team's safety is guaranteed. Naturally, no one is going to take the players to the places where combat activity is under way or there's at least a small hint at an insecure stay.
"There are four [potential locations] at the moment: Kyiv, Lviv, Dnipropetrovsk and Odesa. We are studying the pros and cons of each option and [will] ponder where Shakhtar will be based. But in any case, it will be one of the designated cities."
Palkin added that he was hopeful of being able to confirm the club's location for the coming season in the "nearest future".
"We will immediately inform the public about the decision of our club," he added.
"At the same time, I once again want to draw everyone's attention to the fact that Shakhtar will definitely train and play home games exclusively in the cities recognized by UEFA as safe."