Mr Vijayan was replying in the Kerala Assembly to a motion moved by former State Finance Minister K.M. Mani, who accused the state government of doing nothing to bring back the kidnapped priest.
An angry Mani said: "The state government has a responsibility in the matter. There is no use in just writing letters to the union government. The need of the hour is action from the Centre.
"The hapless priest himself was seen in a video footage saying that he is being kept as a hostage because he is an Indian. We request you (Chief Minister) to please take up the issue on a one-on-one basis with the Prime Minister," said Mani.
Senior legislator (Independent) PC George, backing the issue, said the only way out for his release is to pay the ransom that his abductors are demanding.
Mr Vijayan told the House that he has written three letters to the Centre. "I will now personally take up the issue with the Prime Minister at the earliest," he said
In March 2016 militants barged into a care home for the elderly set up by Mother Teresa in Yemen's Aden and shot dead many people, including four nuns of the Missionaries of Charity, one of whom was from India.
After the shooting, the militants abducted the priest. Since then, besides a few videos, there has been no news of his whereabouts.
Indian officials have said the priest has been in the custody of the ISIS terrorist group in Yemen, but despite efforts at various levels he is yet to be released.
His ancestral home in Ramapuram in Kottayam district is presently closed as two of his brothers live abroad, while another lives in Gujarat.