Gabriel, 25, injured an ankle against Manchester City, leaving them short of defenders for their Premier League opener against Liverpool on Sunday.
Per Mertesacker is out for five months with knee trouble and Laurent Koscielny only returned to training on Monday.
The Gunners are working on a deal to sign Mustafi, 24.
Mustafi, part of the Germany squad that reached the semi-finals of Euro 2016, began his career at Everton but made only one senior appearance for the Toffees, as a substitute in a Europa League match in December 2009.
He joined Sampdoria in 2012 and played 44 league matches for the Italians before moving to Valencia two years later.
Arsenal have made just the one defensive signing this summer - England Under-21 international Rob Holding from Bolton for a fee believed to be about £2m.
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The former Forest Green Rovers striker, 26, has netted 28 times in all competitions for Rovers this season.
His goals have helped the club up to fourth ahead of the season's final day.
"We have an understanding that, if he comes to us and he's got an offer, we'll have a chance to match that offer," Hamer told BBC Radio Bristol.
"If that offer, unfortunately, is so incredibly good for Matty but not very good for Bristol Rovers Football Club, we'd probably have to decline matching it.
"Because, if you're so lopsided, it can de-stabalise the rest of the squad because somebody's on an amount of money.
"He's got an agent that's doing the best job he can for him. But that's not to say we're going to give that one up without fighting."
Hamer also revealed the club would discuss a new contract with manager Darrell Clarke at the end of the season, with Rovers guaranteed at least a play-off spot, lying one point off the top three ahead of Saturday's visit of relegated Dagenham & Redbridge.
"At the moment, we're not distracting him (Clarke) from the job in hand," Hamer said.
"As soon as that's over, we'll sit down and have a chat with him. There's no reason why we can't come up with something that's going to please him and please us.
"The job he has done is remarkable. Having got to know him, I think he could go a long way in this game. He's tactically very aware.
"It would be nice to think he can get to where Eddie Howe is at Bournemouth. There is no reasons why he can't.
"I'd like to think he'll join us on an exciting journey that we've got going forward, but who knows?"
Clarke says Rovers' rivals have "everything to lose" on Saturday and his side would be "very unfortunate" not to go up if they finish on 85 points.

The Dow Jones closed down 122.65 points, or 0.61%, at 19,971.13.
The S&P 500 index dropped 13.8 points, or 0.6%, to 2,280.89, while the Nasdaq fell 47 points, or 0.83%, to 5,613.7.
Airline stocks fell, with American Airlines down 4.4% and United Continental dropping 3.8%.
Michael Hewson, chief market analyst at CMC Markets UK said US markets had taken their lead from weak European markets "as investors wake up to the downside of what a Trump presidency might mean for risky assets".
"The weekend executive order to implement more stringent vetting procedures to visitors to the US from specific destinations, as well as a wider travel ban, could well be the beginning of this realisation.
"It is perhaps no surprise that the tech sector has been the most vocal given that its highly skilled workforce has a higher proportion of overseas workers due to the higher skill set required."
Arsenal defender Gabriel has been ruled out for six to eight weeks, increasing the likelihood of a bid for Valencia centre-back Shkodran Mustafi.
[NEXT_CONCEPT]
Bristol Rovers will have the chance to match offers made by other clubs for League Two's top scorer Matty Taylor this summer, says chairman Steve Hamer.
[NEXT_CONCEPT]
Stocks on Wall Street fell saw their biggest drop so far this year as investors took a cautious stance in the face of uncertainty over Donald Trump's travel and immigration curbs.