The 36-year-old right winger spoke on a conference call from his native Russia on Tuesday, one day after he left the Canadiens to sign a two-year, $10-million deal with the Senators.
Kovalev expressed gratitude to fans in Montreal and tried to clear up misconceptions about his perceived image as a selfish, uncaring player.
"I've been saying many times, I don't know where that comes from," said Kovalev, who has played 16 NHL seasons. "Maybe when coaches say something to me and I don't answer back, they think I don't care.
"All I've been looking for in my career is that I have some kind of communication with my coach."
Problems with communication helped lead to his departure after parts of five seasons with Montreal, where his enigmatic style was revered by fans even if got him in trouble with team management at times.
After Senators general manager Bryan Murray announced he'd signed Kovalev, it was revealed Monday that Canadiens GM Bob Gainey offered the unrestricted free agent an identical deal to stay.
Gainey said when Kovalev hesitated in getting back to him last Wednesday, the opening day of free agency, he went ahead with other deals. The Canadiens acquired Scott Gomez in a trade with the New York Rangers and signed forwards Brian Gionta from New Jersey and Mike Cammalleri from Calgary until there was no money left for Kovalev.
"Everything happened so quick," Kovalev said. "We didn't have much time and they went in the other direction."
Kovalev said his first choice would have been to remain in Montreal so his children, Nikita, 7, and Ivan, 5, wouldn't have to change schools and friends.
"We always choose what's best for our lives and for our families," he said. "When you're 19, it's different. Now, you have a family, it's the first thing you think of.
"That's the only way I look at it. I'm disappointed and feel bad for the kids."
Kovalev was so popular with Canadiens fans that they started an online petition asking Gainey to sign him and make him captain. It garnered 8,547 names by the time he'd signed in Ottawa. About a hundred fans also staged a rally outside of the Bell Centre on Sunday to show their support for bringing back Kovalev, who led the Canadiens with 26 goals and 65 points in 78 games last season.
"I saw the e-mail that a lot of fans sent and I saw the pictures (of the rally), I definitely appreciate it," Kovalev said. "I can't find the words to say to the fans in Montreal. They gave me a lot of support."
Kovalev said he isn't bitter, and won't be looking for revenge now that he is with the Canadiens' Northeast Division rival.
"It's not the first trade for me. It's just part of the life, the way I look at it," he said. "I don't look at it like, 'Oh, they didn't sign me, I'm going to try to do damage."'
After missing the playoffs for the first time in 11 seasons this past spring, Ottawa is faced with its own crisis after star left winger Dany Heatley requested a trade just before last month's NHL draft.
"I think Ottawa is a pretty good team," Kovalev said.
"I just felt that maybe something's missing and maybe I can bring something to help them be competitive and try to win the Stanley Cup," he said. "They have one really good line ... maybe I can be part of giving them a good second line and third line. I'm not a big believer that one line can do the damage on the way to winning a Stanley Cup.
"I'm not a young guy anymore and I'm looking forward to trying to win another Stanley Cup before I retire."
With the Canadiens, Kovalev showed some nights he could be among the most dominant players in the NHL, but was guilty of disappearing on others. He ran into problems with former Montreal coach Guy Carbonneau at times, including being left at home from a road trip in February because of indifferent play.
"I don't know why people say that all the time about inconsistency. I always play the best I can," he said. "Sometimes I try to do too much because I feel I can try to change the game. Sometimes it works and sometimes not, but I always play my hardest."
With the Senators, Kovalev won't be expected to carry the offensive load as much as he was in Montreal. Even if Heatley doesn't return, he'll get a chance to play with leading forwards Jason Spezza and Daniel Alfredsson.
"I'm not trying to be the superstar and try to be a hero," he said. "I'm just trying to do my best to help the team win the Stanley Cup."