The 25-year-old has signed a three-year contract, and although he has yet to collect a visa and work permit, it is hoped he will be available for Saturday's Barclays Premier League clash with Everton at Ewood Park.
"Hopefully he'll be back in the country some time tomorrow [Wednesday]," said Kean on Tuesday, whose side are at home to Sheffield Wednesday in a Carling Cup second-round tie on Wednesday.
"It's going to be difficult for him to be involved in (the Carling Cup) game because he has not even trained with us.
"But if he comes back with all the appropriate paperwork then he will train with us on Thursday and Friday in preparation for the Everton game."
It is understood Rovers have paid Sporting £2million for Vukcevic, who is his country's most-capped player.
Assessing Vukcevic, Kean added: "He is an exciting player who can play wide on either side.
"As a character, when you meet him he is a very focused guy, takes his football very seriously. He likes to take people on, to be the guy who makes a difference, a bit of a match winner.
"He is a very good player with lots of international games under his belt for a country in the top 20, so he has performed at that level at a consistent basis.
"He will give us a different dimension."
In revealing it had not worked out well for Vukcevic towards the end of his time at Sporting, Kean said: "So he has decided to come to us, which is brilliant for us.
"I've watched him a number of times live and if you can get the ball out wide to him and he can isolate a full-back, then it will be hard for anyone to stop him when he is on the run.
"That's exactly the type of player we have been after."
Kean has also confirmed he is close to signing centre-back George John from Major League Soccer side FC Dallas.
Rovers look set to pay a fee for John of around £1.2million to the MLS, who hold the player's contract.
Personal terms are close to being finalised, with Rovers finally poised to fill the gap vacated by Phil Jones' £16.5million move to Manchester United.
To avoid the necessity of obtaining a work permit, John has travelled to Athens to collect a passport, with the 24-year-old holding dual citizenship as his grandparents were born in Greece.
"All the paperwork with the MLS has been agreed, the player has travelled to Greece to get his passport, and hopefully he will be in the country either tonight or tomorrow[Wednesday]," said Kean.
"We've been tracking him through a scout based in the USA, we have had a number of good reports, and I've seen him myself on a number of occasions.
"In times past I've been involved with a lot of American players - Eddie Lewis, Marcus Hahnemann, Clint Dempsey, Brian McBride - from my Fulham days.
"It's a very good market because you can get players who have great potential, who hit the ground running, and you can normally get them for a very competitive salary, and he fits the ball.
"He is 6'4", a commanding defender, he is quick, aggressive, and he has a really bright future."
Kean is due to fly out of the country on Tuesday night in a bid to sign a new striker, although claims it is not Bosnian Nenad Ibisevic.
Another player also interesting Kean is midfielder Jimmy Bullard, whose contract is in the process of being terminated by Hull following an incident on a pre-season trip to Slovenia.
"I've probably been linked with Jimmy because I met him on Sunday when he was out with his boy, and I had a chat with him and he was telling me about his troubles, and he could be available," said Kean.
"I had Bully when I was with Chris Coleman at Fulham, and he is a colourful character.
"But I don't know how his fitness is, whether he has done a full pre-season. But he is a quality player when he is on his game.
"It depends if we need somebody in that position, and if we have done our business and filled those slots, then it might not happen, but if we haven't, then you never know."