No fee was announced but media reports said the 20-year-old had been signed from Benfica for £20 million.
"I am very happy that I have joined such a great club," Markovic said on the club website.
"I hope that I will fulfil the expectations of the supporters, coaches, players and all connected to the club. I have seen what the players are like, I have seen what the supporters are like, and I cannot wait to start playing here."
Belgium striker Divock Origi is also close to joining Liverpool in a £10 million deal from Lille, according to media reports, as manager Brendan Rodgers bolsters his squad after the departure of Uruguay forward Luis Suarez to Barcelona.
A former Partizan Belgrade player, Markovic helped Benfica win the Portuguese title last season after joining the club in June 2013.
"This is an exciting signing for us and one that improves our squad, in terms of attacking options, for the coming season," Rodgers said.
"He was hugely impressive for Benfica last season and has proved he has the ability to produce the goods in European competition as well, which will be important for us."
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