The two-time Stanley Cup champion wrapped up a seven-year, $42 million contract that paid him $5 million this season, though his cap hit was $6 million.
It's believed Elias is looking for a multi-year deal worth around the same salary he earned this year.
Elias, 37, led the Devils in scoring this past season, registering 14 goals and 22 assists for 36 points.
The native of Trebic, Czech Republic, has played in 1,090 career NHL regular-season games, all with the Devils, and has accumulated 375 goals and 555 assists for 930 points, along with a plus-191 rating and 497 penalty minutes.
He was originally drafted by Devils in the second-round (51st overall) of the 1994 NHL Entry Draft.
Meanwhile, the Devils are not expected to re-sign forward David Clarkson, who has fielded several calls from numerous teams today.
Clarkson, 29, could command a salary north of $6 million per year on the open market.
In 48 games with the Devils this past season, Clarkson registered 15 goals and nine assists for 24 points, along with 78 penalty minutes.
The Toronto native has played in 426 career NHL regular-season games, all with the Devils, and has accumulated 97 goals and 73 assists for 170 points, and 770 penalty minutes.