The new Bluebirds boss is also part of Chris Coleman's Wales coaching team.
Trollope was appointed on 18 May and flew to Malaysia for his first meeting with Tan and will rejoin the national squad in Portugal later in the week.
Meanwhile, Cardiff have announced they will play Borussia Mönchengladbach in a pre-season tournament in July.
The Bluebirds will FC St. Pauli, and VfL Osnabrück as well as Borussia, who finished fourth in the 2015-16 Bundesliga.
Trollope will stay with Wales throughout the European Championship, but Coleman has believes Trollope is unlikely to be able to do both jobs after the tournament.
BBC Sport has learned that the first meeting between Trollope and the sometimes controversial Tan was positive as the new head coach prepares to put his own backroom staff together at the Cardiff City Stadium.
Paul Trollope missed the start of Wales' pre-Euro 2016 training camp to have his first meeting with Cardiff City owner Vincent Tan.