A-Rod is now a free agent after he was released by the Yankees on Saturday. Hill said the 41-year-old clears waivers on Monday, and the club may at that time reach out to the 14-time All-Star.
"We'll look at everything," Hill said. "There's been information out there about [Rodriguez's] situation."
Even before Stanton suffered a groin injury on the last play of Saturday's 8-7 loss to the White Sox, Miami was open to signing Rodriguez as a first base option.
After Sunday's 5-4 win, the Marlins announced Stanton has a Grade 3 groin strain and will miss a minimum of six weeks. So there is an outside chance he returns at the end of the year.
The possibility of adding Rodriguez, who has 696 career home runs, could be enticing for the Marlins, because it would be a low-risk, high-reward move.
One thing the Marlins want more clarity on is the personal service contract Rodriguez has agreed to become an advisor for the Yankees in 2017.
"I have no idea what his interest level is to continue playing," Hill said. "He has something set up to move into the [Yankees'] front office. I don't know where his head is on that, but where we are now, this team has played too well for too long. We're right in the thick of this thing, and we're going to do everything in our power to help in any way we can. If it's him, or if it's someone else, we're going to try to find a way."
Along with Stanton, the Marlins also have first baseman Justin Bour on the disabled list with a right high ankle sprain.
"Injuries are part of the game," Hill said. "They're something we can't control. You know things are going to happen."
Stanton (25) and Bour (15) had combined for 40 home runs. They've also driven in 116 runs -- Stanton (70) and Bour (46).
Manager Don Mattingly stated bluntly that others have to step up.
"It means we've got to find another way to win, simple as that," Mattingly said. "We did it early in the year. Giancarlo early in the year was not really contributing and he wasn't swinging well, but we were able to sustain. I think that's really what we have to do. We talked about it in Spring Training. We have to find a way to win games."
Rodriguez lives in Miami, and mumblings have already begun that if he signed with the Marlins, he could take his daughters to school before going to the ballpark.
One catch is finding a place for Rodriguez to play. First base is the likely choice with Bour out.
Stanton is being replaced in right field by Ichiro Suzuki.
But Rodriguez has appeared in just one game at first base, and that was in 2015.
Still, the longtime shortstop and third baseman has the athleticism to handle first base, according to Mattingly.
"He moved over to third, almost seamlessly," Mattingly said. "There's no reason he couldn't play first."
The Marlins currently are going with Chris Johnson against left-handed starters, and Derek Dietrich when a right-hander is on the mound. Utility infielder Miguel Rojas has played some at first, but he's more of a defensive specialist.
Bour's power has been missed at first base. The Marlins haven't had a home run by the first base spot since Johnson went deep on July 20.
And for the season, Johnson has two of his four homers when he's started at first base.
Dietrich has made 10 starts at first base, and he has yet to homer or driven in a run from the position.
If Rodriguez isn't the answer, the Marlins plan to still explore candidates who might be available before the Aug. 31 deadline for a player to be eligible for the postseason roster. Outfielder Carlos Gomez, designated by the Astros, also is under consideration.
"Nobody is going to feel sorry for us," Mattingly said. "Everybody has injuries. Every team that I've ever been involved with has things that they have to overcome. This is just one of them. We've got to do what we've got to do."