The navy said the ships had withdrawn for "prior operational reasons", although a port official indicated they had been due to take part in an exhibition on Wednesday.
"They had come there as part of preparations for the navy week exercise and moved out of the port for some prior operational reasons," spokesperson DK Sharma told AFP.
"Like every year, the ships will return as and when the opportunity exists."
The move came as police and the Indian Coast Guard said the city and port had been placed on high alert for a possible attack.
"We have received a fax on Tuesday afternoon from central intelligence agencies warning of the possibility of a militant attack in Kolkata, especially in the port area," senior Indian Coast Guard official BN Mahato told AFP.
"We have deployed hovercraft to intensify (a) night-long vigil in the Bay of Bengal."
Kolkata Police Commissioner Surojit Karpurakayastha said security had been stepped up in the port area and parts of the city.
Police were "acting on the basis of the central intelligence agencies' alert," he said.
No further details about the alert were immediately available, but it came days after a suicide bomb attack near the Pakistani side of the Wagah border that killed 55 people and wounded more than 120.
India placed several states on alert in September, when al Qaeda announced it was launching a new branch to "wage jihad" in South Asia.
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