Expectation was increasing that Albion could bring in a World Cup centre-back amid reports of an imminent deal.
Updates from Italy and Germany on Saturday evening suggested the Seagulls have made their move for Venezia skipper Michael Svoboda, who has a transfer release clause of almost £5 million.
He is currently involved at the World Cup with Austria but did not feature in their opening match against Jordan.
They play Lionel Messi and Argentina on Monday.
The 6ft 5in 27-year-old led Venezia to the Serie B title last season.
Albion are facing a busy season of domestic and European commitments and, in terms of central defenders, have released Adam Webster as well as selling Jan Paul van Hecke to Spurs.
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They are believed to have bid in excess of £40 million for Luka Vuskovic, the Croatia defender who is currently with Spurs but does not feature in Roberto De Zerbi's immediate first team plans.
Meanwhile Olivier Boscagli and Diego Coppola are likely to attract interest from Europe.
They have so far been linked with Benfica and Lazio respectively.
Venezia are also going into a busy summer after bringing in significant new investment as they look to become more than occasional single-season visitors to Serie A.
The one previous piece of business between the two clubs was four years ago when striker Aaron Connolly joined Venezia on loan for the Serie B season.
The deal was cut short midway through the campaign.
Head coach Fabian Hurzeler said Albion are "always" ready to act in the transfer market.
The Seagulls boss told TalkSPORT: "It's really important that we make the right signings that makes sense for us.
"That fits the characters we already have, that fits to the style of play from us.
“So, let's try to make a rational, good decision.”
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