"If it comes back and he’s guilty, then we’ll move forward with the execution," Rep. Kevin McDugle, a Republican, said at a press conference on Tuesday.
"If it comes back and he’s innocent, then we’ll definitely do everything in our power to stand up and be as loud as we can to make sure he has another opportunity to walk free."
Glossip was sentenced to death after he was convicted of ordering the murder of Oklahoma City motel owner Barry Van Treese in 1997.
The man who beat Treese to death, Justin Sneed, admitted to the murder and was sentenced to life in prison, but said he only did it after Glossip promised to pay him $10,000.
Glossip’s attorney, Don Knight, claims that Sneed only implicated Glossip in the scheme to avoid the death penalty himself.
Glossip was set to be executed in October 2015 but it was postponed at the last minute when prison officials realized that they had the wrong drugs for lethal injection.
Oklahoma put a moratorium on executions after a series of flawed lethal injections in 2014 and 2015, but that ended when 60-year-old John Marion Grant was put to death last October.
Oklahoma Attorney General John O'Connor noted that Glossip has been tried and convicted twice, saying that if his defense has uncovered new evidence, then they should present it to the Court of Criminal Appeals.
"Mr. Van Treese was murdered on January 7, 1997. Glossip has had 24 years of appeals," O'Connor said in a statement. "It is time for justice to be served for Mr. Van Treese’s family and the people of Oklahoma."
Stan Perry, an attorney with Houston-based law firm ReedSmith, said that 20 attorneys volunteered to take part in the investigation into Glossip's case.
"It’s a diverse group of lawyers, backgrounds, experienced prosecutors to civil litigators, all of who are giving their time and realize this is important," Perry said Tuesday. "It’s our hope that the results of the investigation, whatever those results are, allow closure for all sides."
Once the investigation is complete, the results will likely be presented to the pardon and parole board, which could recommend that the governor pardons Glossip or commutes his sentence.
Oklahoma County District Attorney David Prater could not be reached for comment on Tuesday.
The Associated Press contributed to this report.