The Spaniard impressed at Leipzig, but the German club have been unable to finance their €30m option to buy.
Julian Nagelsmann recently said he was optimistic of retaining Angelino, but Leipzig want another loan deal.
However, the new interest from Barcelona could change the outlook of Angelino’s future.
Unlike Garcia, Angelino has three years remaining on his contract, putting City in a stronger negotiating position.
Blues value Angelino at €30m
City have informed Barcelona that they want €30m (£26.5m) to sell Angelino, a former Spain under-21 international.
No official offer has arrived yet and given their financial situation, Barca will likely to have sell before considering such an investment.
Angelino joined City from the Deportivo La Coruña academy in 2013 and worked his way up through the ranks.
Loan spells at New York City FC, Girona, Real Mallorca and NAC Breda followed, before PSV signed him permanently.
His good form at the Eindhoven club led to City using their buy-back clause after just one year.
Angelino appeared 12 times for Pep Guardiola’s side, but went to Leipzig in search or more regular football.
In Germany, the wing-back flourished in his 18 appearances, delivering four assists in the Bundesliga and Champions League.