The tiny nation will qualify if they win in Budapest and Serbia slip up against Bulgaria; meanwhile, their hosts have already sealed a spot in Germany.
Match preview
Securing their place in a third straight European Championship finals, Thursday's dramatic draw against Bulgaria was enough to edge Hungary over the line and ensure a top-two finish in Group G.
The Magyars strode out in Sofia - where the game was played behind closed doors, amid violent protests outside - knowing that qualification was within touching distance, and a late turn of events ultimately booked their ticket.
Trailing 2-1 at the end of 90 minutes - and down to 10 men following a red card for Milos Kerkez midway through the second half - Marco Rossi's men struck again in the seventh minute of stoppage time, as an own goal by Bulgaria's Aleks Petkov saw the spoils shared.
Thereby extending their unbeaten run to 11 matches across all competitions, Hungary have picked up 15 points from a possible 21 to seize the summit of Group G, now sitting two points above second-placed Serbia and four clear of Sunday's opponents.
Having previously hung on for a crucial 2-1 home win over the Serbians, before coming from two goals down to rescue a draw in Lithuania, there is no doubt Rossi's side boast deep reserves of character.
They will now try to sign off their qualifying campaign in style this weekend, before preparations for next year's Euros begin in earnest.
Despite having less than one-third of possession, Montenegro held Hungary goalless in June's reverse fixture, before taking four points from games against Bulgaria and Lithuania to keep themselves in top-two contention.
However, the side coached by Miodrag Radulovic were dealt a major blow by suffering a painful 3-1 loss to bitter rivals Serbia last month; leaving them in danger of falling before the final hurdle.
Victory was essential, then, as the Brave Falcons played host to Lithuania in their penultimate Group G fixture on Thursday evening, when Edvin Kuc's early goal eased the nerves at Podgorica City Stadium and captain Stevan Jovetic sealed the deal not long after half-time.
Having posted three successive home wins for the first time in over a decade, Montenegro now travel to Budapest in the knowledge that they can still qualify directly for their first finals since achieving independence back in 2006.
Not currently pencilled in for a playoff place, the little nation of around 600,000 people must upset Hungary and rely on Bulgaria to beat Serbia, so snatching second place remains something of a long shot.
Hungary Euro Champ Qualifying form:
D
W
W
W
D
D
Hungary form (all competitions):
W
W
D
W
D
D
Montenegro Euro Champ Qualifying form:
L
D
D
W
L
W
Montenegro form (all competitions):
L
D
W
W
L
W
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Team News
Due to his dismissal in midweek, Milos Kerkez will serve a suspension on Sunday, so Marco Rossi must make at least one change to Hungary's starting XI at Puskas Arena.
Rossi's options are also restricted by the absence of Willi Orban, Attila Fiola, Laszlo Kleinheisler and Roland Sallai, all of whom are unavailable this month.
Nonetheless, Martin Adam should continue to lead the line up front after scoring the opener against Bulgaria, with Liverpool star Dominik Szoboszlai supporting the Ulsan Hyundai striker from an advanced midfield role.
Meanwhile, Montenegro's attack will be led by Stevan Jovetic, who is the Brave Falcons' all-time leading scorer with 34 goals, notched at a strike-rate approaching one every two games.
The visitors' skipper should be partnered by Lecce's Nikola Krstovic, who has intermittently impressed since moving to Serie A in the summer and has scored twice in seven Euro 2024 qualifiers to date.
Another forward, South Korea-based Stefan Mugosa, was forced to withdraw from this month's squad due to injury, in addition to Lazio full-back Adam Marusic.
Hungary possible starting lineup:
Dibusz; Balogh, Lang, A. Szalai; Nego, A. Nagy, Styles, Z. Nagy; Szoboszlai; Adam, Nemeth
Montenegro possible starting lineup:
Mijatovic; Vesovic, Savic, Tomasevic, Radunovic; Osmajic, Kuc, Jankovic, Camaj; Jovetic, Krstovic