The appeal is scheduled to be heard Wednesday with a decision to follow on whether Meriweather will be permitted to play in Sunday's game against the Denver Broncos.
Meriweather was suspended for two games Monday after yet another Sunday marked by personal fouls. If his suspension stands, it will cost him $141,176.
Meriweather was flagged twice for personal fouls in the Redskins' 45-41 win over the Bears. He was called for a helmet-to-helmet hit on receiver Alshon Jeffery at the end of a 28-yard catch late in the third quarter. Late in the fourth quarter, Meriweather launched into receiver Brandon Marshall after he dropped a pass in the end zone.
After the game Marshall said he thought the league should consider kicking out Meriweather because of the way he plays. Many Bears weren't happy Tuesday either.
"I still want to punch him in the face," tight end Martellus Bennett said during his weekly radio show on WSR-670 in Chicago.
Informed of the league's suspension of Meriweather, Bennett quipped: "We don't get any of his money."
Marshall spoke with ESPN's Linda Cohn on Tuesday and said he understands that rule changes have made it tougher on defensive players but that they are the rules and they've been implemented for a reason.
He also said that he reached out to Meriweather to discuss the situation but came away without much agreement.
"It was one of those things it seemed like he was more concerned about being suspended and losing some money when I was just trying to explain to him (to) forget that, let's talk about life," Marshall said. "Let's talk about life after football, being able to function. And trying to express to him that you're hurting yourself too physically."
It's not the first time the NFL has punished Meriweather. He was fined $42,000 for two hits in a Week 2 loss at Green Bay. And he was fined $50,000 for a hit in 2010 with New England and accrued $45,000 in fines a year later with Chicago.
There is precedent for Meriweather winning an appeal. Tampa Bay safety Dashon Goldson had his one-game suspension overturned earlier this season after a helmet-to-helmet hit on the New Orleans Saints' Darren Sproles, although he was still fined $100,000. Last season, Baltimore Ravens safety Ed Reed won his appeal and his one-game suspension for a hit to the Pittsburgh Steelers' Emmanuel Sanders was reduced to a $50,000 fine.
Washington's other starting safety, Reed Doughty, exited Sunday's win with a concussion. The Redskins have no proven starter behind Doughty and Meriweather, and have been forced to often play three corners and one safety.
ESPN.com Redskins reporter John Keim contributed to this report.