The juveniles escaped from Subhayan -- a government-run home in Balurghat in South Dinajpur district, 440 km from Kolkata, police said.
"We have been able to apprehend 19 of them. We are looking for the other two," Superintendent of Police (South Dinajpur) Prasoon Bandyopadhyay said.
The 19 juveniles were apprehended close to the India-Bangladesh border, he said.
He said the delinquents were mostly illegal Bangladeshi immigrants.
There have been previous instances of children running away from the care home, which now has 33 juveniles. Ten Bangladeshi children had fled from the home in February 2010.