The Scot was keeping an eye on the pair as Pep Guardiola's side picked up another three points thanks to goals from Mario Gotze and Thomas Muller
Earlier this week, the Brazilian's agent claimed the German champions were "in a race" to sort a contract extension and that United's interest in the 30-year-old has been long-standing.
Kroos, meanwhile, has seen talks over a new deal at Bayern stall, with chief executive Karl-Heinz Rummenigge admitting the club is keen to extend his contract which expires next year.
Bayern Munich resumed its Bundesliga campaign with a 2-0 win over Borussia Monchengladbach. Mario Gotze struck after just seven minutes before Thomas Muller's penalty wrapped up the victory for Pep Guardiola's side, which is 10 points clear at the top of the table.
Moyes already traveled to Italy earlier this month to watch Juventus midfielder Claudio Marchisio as well as Cagliari's 27-year-old center back Davide Astori.
Meanwhile, United confirmed on Friday evening that it hs agreed a club-record 37 million pound fee for Juan Mata, with the Spain international set to undergo a medical and complete personal terms.
Mata will need to be registered by Monday lunchtime in order to make his debut against Cardiff City on Tuesday.