The Dane - who suffered a heart attack at the Euros last year - has been a revelation for Brentford since signing in January.
Eriksen is out of contract in the summer, but Frank says Brentford will attempt to keep him in west London.
Speaking after a 3-1 loss to Manchester United on Monday night, Frank said: "Eriksen is again a very important player for us.
"The key is he performs consistently at a high level. It is no secret we would like to do everything we can to keep him."
On the result, Frank added: "We played a Manchester United team who played well today, so congratulations to them.
"I think we started well for the first five minutes but then Manchester United were on top. Second half I saw much more my team I want to see play. We were back in the game but made a stupid penalty. It took a little bit out of everything and then we conceded on a set piece. Better second half.
"We were in the game [at 1-0]. We had opportunities and we were equally in the game, which was good.
"I don't think it was a 3-0 defeat in terms of chances. I think it was a fair loss. We have had a very good season in many ways. We win or learn and today we learned a lot."
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