"We haven't lost games because of our goaltending," Sullivan said Tuesday.
Sullivan, though, also hasn't outright declared Murray will start Game 5 after allowing eight goals on 58 shots in losing back-to-back games at Bridgestone Arena, including three on 25 shots in a 4-1 loss in Game 4 on Monday.
That leaves open the possibility, even if it's slim, Fleury could start Game 5 at PPG Paints Arena on Thursday (8 p.m. ET; NBC, CBC, SN, TVA Sports) with the best-of-7 series tied 2-2.
Video: PIT@NSH, Gm4: Murray stones Neal twice with his pad
Fleury hasn't played since Game 3 of the Eastern Conference Final, when he allowed four goals on nine shots in 12:52 before Murray replaced him. Going into Game 3, Murray was 5-1 with a 1.54 goals-against average and .943 save percentage in the 2017 Stanley Cup Playoffs.
In his past two games, Murray's GAA has gone up to 2.08 and his save percentage has dropped to .925.
"Like anything, we make our lineup decisions on a game-by-game basis," Sullivan said. "The one comment I will make is we didn't lose the game [Monday] night because of our goaltending."
But you can make the case the Predators won Game 4 because of their goaltending.
Pekka Rinne was outstanding with 23 saves on 24 shots, many from directly in front of him, including some off breakaways or chances generated after the initial save on a breakaway.
If Rinne is going to be that good, the Penguins are going to need their goaltending to at least match him. Murray didn't do that, but he didn't get much help either.
Sullivan, in fact, traced that back to the past two games, when he said there were some goals the Penguins allowed that were preventable. There is no debating him on that, and it's a reason why Murray deserves a partial pass.
Predators defenseman Roman Josi scored a power-play goal at 5:51 of the second period in Game 3 on a shot deflected by Penguins forward Carter Rowney's left hand; Rowney was too late getting into the shooting lane to make the block.
Video: OTT@PIT, Gm2: Fleury shuts down Dzingel in close
Nashville forward Frederick Gaudreau scored 42 seconds later because he got behind Pittsburgh defenseman Trevor Daley, forcing his partner, Ian Cole, to back up, giving up his gap and space to Gaudreau, who beat Murray with a glove-side shot from between the circles.
Then there was Predators forward Craig Smith scoring on a breakaway off a Penguins turnover, a mix-up between forwards Chris Kunitz and Phil Kessel, at 4:54 of the third period.
Forward Calle Jarnkrok got the scoring started for the Predators in Game 4 with a goal off a rebound at 14:51 of the first period. The play wouldn't have happened if Penguins cleared the puck out of the zone. They had a chance to do so and couldn't get it done.
Nashville forward Viktor Arvidsson scored on a breakaway at 13:08 of the second period after he got in behind Pittsburgh forward Patric Hornqvist and defenseman Justin Schultz. The Penguins didn't have good puck coverage in the neutral zone either.
"There are some goals that are scored because good players make good plays, and you're playing against a good team," Sullivan said. "Sometimes that happens. Then there are other circumstances where goals are scored where, as a team or a coaching staff, we feel we can do a better job in avoiding those situations. There were one or two of them in the game [Monday] night. We think they were preventable goals if we make certain reads in those situations."
Sullivan pointed out the Predators were thinking along those same lines after Game 2, when they didn't allow many chances but still gave up nine goals and lost each game.
That led to questions about Rinne's status as Nashville's goalie for Game 3. Like Sullivan now, Predators coach Peter Laviolette refused to outright say Rinne was starting, but he never quite gave the impression he was thinking of making a change.
Video: Discussing the Pittsburgh Penguins in Game 4
However, because Laviolette didn't say anything definitive, the Rinne situation led to a media-manufactured talking point in the series, much in the way the Murray situation appears to be heading now.
Rinne knew the whole time he was starting Game 3. Maybe Murray already knows he's going to start Game 5. Or maybe not. It wouldn't be unprecedented for Sullivan to make a change.
Not only did he change from Fleury to Murray earlier in this postseason, last year he changed from Murray to Fleury for Game 5 of the Eastern Conference Final against the Tampa Bay Lightning.
The Penguins lost Game 4 and the series was tied 2-2, like the Cup Final is now. Fleury started Game 5. He made 21 saves, but Pittsburgh lost 4-3 in overtime.
Sullivan went back to Murray for Game 6. He won the next two games and then four out of six in the Stanley Cup Final against the San Jose Sharks.
"We think one of the strengths of our team is in our goaltending position," Sullivan said.
The question is which goalie does Sullivan think gives the Penguins the best chance to win Game 5? It's probably Murray.