The 23-year-old, who currently plays for Benfica, is a versatile attacker who usually operates in the No.10 role but can also play elsewhere.
Norwich City are also interested in the playmaker, but Albion are currently in pole position.
The Baggies are planning to approach Benfica with an offer, although the Portuguese club may prefer a loan deal because Krovinovic still has three years left on his deal.
Albion are keen to lower the average age of their squad this season and Bilic has identified the Croatian under-21 international as a key target for the summer.
After Bilic’s first week in charge, transfer activity is now beginning to gather pace.
Craig Dawson is expected to leave for Watford on Monday in a £6million deal to Watford.
However, the Baggies are not interested in former Aston Villa defender Alan Hutton, as reported elsewhere.
The 34-year-old is not in their preferred age group as they look to freshen up their squad with younger players on lower wages.
Krovinovic fits the bill, and Bilic’s proposal to chase him has been warmly received by the board.
Born in Zagreb, he played 78 games for hometown club NK Zagreb before joining Rio Ave in Portugal in 2015.
He made 50 appearances for them across two seasons before Benfica signed him in 2017.
He was a regular in his first season for Benfica before injury scuppered his chances and he struggled to get back into the team last season.
The Baggies are keen to field an energetic team next season, and could also be boosted by a potential u-turn from Rekeem Harper.
The 19-year-old is out of contract tomorrow, but he trained all last week with Albion and could now sign the four-year deal that has been on the table since the end of the season.