Constantine, under whom India lifted the LG Cup in Vietnam -- a rare international title win, is considered to be the front runner to succeed Theodore Whitmore as head coach of Jamaica's Reggae Boyz ahead of their match against Panama in September for 2014 World Cup qualification.
Constantine was the main lecturer in Jamaica Football Federation's Advanced Level Two coaching course last month and a 'Jamaica Observer' report said that he is interested in the job and is being seriously considered.
The 50-year-old Cypriot, who was with the Indian team from 2002 to 2005, has a wealth of experience as a coach, having also managed the national sides of Nepal (1999-2001), Malawi (2007-08) and Sudan (2009).
He has been first team coach of the English Championship side Millwall for the 2005-06 season. Most recently, he managed Nea Salamis in the Cypriot First Division, after guiding them to promotion from the Cypriot Second Division.
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India won the six-nation LG Cup in August 2002 after beating home side Vietnam 3-2 in the final with then captain Bhaichung Bhutia leading the charge.
Jamaica has been without a head coach after Whitmore opted to resign following the Reggae Boyz 0-2 defeat to Honduras which left them with a slim chance of qualifying for the 2014 World Cup in Brazil.
Jamaica are currently at the bottom of the CONCACAF six-team group on two points from six games and with four matches remaining, but still have an outside chance of finishing in the top four.