Oilers GM Craig MacTavish would not reveal who he has brought in as coach, but indicated that the club is "very close" to an agreement.
Sources have told TFP that Eakins has agreed to terms on a deal, but has not yet signed his contract.
The Vancouver Canucks, Dallas Stars and New York Rangers were also courting Eakins.
Krueger, named the 11th coach in club history on June 27, 2012, registered a 19-22-7 record this season. Prior to being named head coach, he spent the two previous years as the Associate Coach of the Oilers.
The Winnipeg native served as the Head Coach of the Swiss National Team from 1997-2010. He led the team to 12 World Championships and three Olympics and moved the country up eight spots in the IIHF world rankings, from 15th to 7th during his tenure.
Eakins wrapped up his fourth year as head coach of the Toronto Marlies of the AHL. He was named the third head coach in team history on Aug. 4, 2009. He previously served as the Director of Player Development for the Toronto Maple Leafs.
Eakins, 46, also served as an assistant coach for the Maple Leafs from 2006 to 2008.
The Dade City, Florida native was originally drafted by the Washington Capitals in the 10th round, 208th overall, in the 1985 NHL Entry Draft. Eakins played in 120 games for Winnipeg, Florida, St. Louis, Phoenix, the New York Rangers, Toronto, the New York Islanders and Calgary, collecting nine assists and 208 penalty minutes. He retired from professional hockey following the 2004 season with the Manitoba Moose of the AHL.