Garcia is out of contract at the Etihad next summer and has expressed a desire to return to Barca, the club he left in 2017.
Koeman would also like to be reunited with Lyon's Memphis Depay, who he worked with during his time in charge of the Dutch national team, although he conceded Barca's financial situation is holding them back in the transfer market.
"There are two positions in the team where signings would do us good," Koeman told a news conference on Saturday. "We need a centre-back and a No. 9. These are two positions where it would help to have more bodies.
"We've got until Monday. It's true we want [Garcia] here. We [only] have three or four centre-backs, so it's a bit tight.
"I hope we can get it done, but we know the situation is tough financially. There's interest but we don't know if we can do it."
Koeman did not name Depay directly but sources have told ESPN that he has emerged as a low-cost alternative to Inter Milan's Lautaro Martinez due to his contract situation at Lyon. Like Garcia, he's into the final year of his contract.
Signing Depay could be dependent on first selling Ousmane Dembele. Manchester United have turned their attention to the France international as they continue to be frustrated in their pursuit of Borussia Dortmund's Jadon Sancho.
Dembele, 23, has been reluctant to leave Camp Nou in the past but sources have told ESPN international teammate Paul Pogba has encouraged him to move to Old Trafford.
Barca would be open to selling but are reluctant to let him leave on loan, while Koeman said Dembele is free to decide his own future before Monday's transfer deadline.
"Dembele is a Barcelona player," he added. "[His future] depends on what he's thinking. I have not spoken to him about it. We will see.
"Ansu's ahead of Dembele. If a player's not happy, they can speak to me. Ousmane's not said anything to me. If he stays, I'm counting on him and he will get chances to play."
Dembele is expected to be included in the squad for Sunday's game against Sevilla at Camp Nou, when Barca look to continue their excellent start under Koeman by making it three wins from three in La Liga.
"If the papers are through, he will be in the squad," Koeman said. "I see him as a starter [this season], but there is competition. [Sergi] Roberto's doing well, we're winning and the players coming in need to provide competition.
"For sure, Dest's the future, but that doesn't mean he can't play straight away because of the quality of the player. If he needs to play tomorrow, I am not afraid about it."