"Turkey will always be a sincere partner of dialog for Lithuania," Linkevičius told reporters after meeting with his Turkish counterpart, Mevlut Cavusoglu, in Vilnius.
"It's abnormal not to speak directly, even when there are problematic issues, whether it is human rights or detentions of people," the Lithuanian minister said.
"We'll do our best to bring about dialog with the European Union and we are trying to make that dialogue in the spirit of trust," he added.
Relations between the EU and Ankara took a turn for the worse after thousands of Turkish public servants and officials were arrested in the wake of a failed coup attempt in 2016.
Brussels accuses Turkey of human rights violations, but Ankara says that the EU failed to show solidarity.
"Unfortunately, our EU friends didn't want to understand what we faced that night and afterwards," Cavusoglu said in Vilnius.
Linkevičius was the first EU and NATO foreign minister to visit Turkey after the failed coup.
The EU said last June that Turkey's EU accession talks had "effectively come to a standstill".
Cavusoglu said, however, that Turkey would undertake reforms after it lifted its state of emergency in July.
"We still want to become an EU member despite the difficulties", he said, thanking Lithuania for support.
Ankara 'doesn't have the luxury to choose'
Cavusoglu also said that Turkey did not see its partnership with Russia as an alternative to its relations with Lithuania or the West.
Ankara's relations with Moscow have been warming in recent years. Turkey plans to purchase S-400 air defense systems from Russia. The presidents of Russia, Turkey and Iran are to hold a summit on Syria next week.
There have been tensions in Ankara's relations not only with the EU, but also with Washington. A trade standoff between Turkey and the US sent the lira down by almost 20 percent this month and military cooperation between the two NATO allies may weaken, too.
The Turkish foreign minister said that Ankara did not have "the luxury to choose" partners, but it managed to "perfectly balance" its foreign policies.
"Our relations with Russia is not an alternative to our relations with Lithuania, Europe or the European Union, (...) or United States, even though we have some problems," Cavusoglu told reporters in Vilnius.
"We have been waiting at the EU doors for 16 years and we are still there, but Turkey or any country like Turkey living in this part of the world shouldn't prefer between this country and that country," he said.
Linkevičius said he saw no reason to think that Ankara's agreements with Russia might go against Lithuania's interests.
"That won't happen. I'm sure about that," the Lithuanian minister said.