About 2,600 troops of the former Soviet armed forces, mainly officers, remain in Estonia.
"We cannot take sole responsibility in building for all these officers who leave Estonia," Luik told Swedish state television news after meeting his Swedish opposite number.
"We are ready to help, but we don't want ourselves to be blackmailed," he added, saying his country was too small to afford full rehousing payments.
Talks on the troop withdrawal issue have been plagued by problems since the first Russian pledge to pull out over 18 months ago.
Luik said Russia's delegation told Wednesday's meeting Moscow could no longer stand by a jointly agreed date for completion of the pullout by Aug. 31.
It was not clear where the talks had been taking place.
"We were nearly there," he was quoted as saying by the national news agency TT. "The agreement was on the table and all the details were completed."
Asked if the development implied that there was Russian resistance to completing the pullout as agreed, Luik told television: "We have felt for the last months that there is a certain Russian desire or interest ... not to withdraw them as quickly as possible."
Luik repeated a call for international support for Tallin's position, telling the television news: "We need strong international pressure and commitment so the troops (will) still leave Estonia."
Estonia appears to be having the most difficulty of the three Baltic states, which include Latvia and Lithuania, in completing the withdrawal talks.
In 1993, Russia withdrew all its troops from Lithuania. Latvia, keen to get up to 20,000 troops off its territory by August, recently reached a deal with Russia over the Skundra radar base which had been a major stumbling block.
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