Ireland apologized for the question, and the NFL players union raised concerns Wednesday about discrimination and degradation.
Ross issued a statement saying he'll look into the matter personally.
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"As an owner of many companies and organizations, including the Miami Dolphins, I have always strived to comply with the highest standards in all aspects of my businesses, including recruiting," Ross said. "In interviewing employees, we always look to obtain relevant and appropriate information in adherence with the best industry practices."
An NFL statement said Ireland "exercised poor judgment in asking an insensitive and inappropriate question."
League spokesman Greg Aiello added, "Steve Ross' statement makes clear that the Miami Dolphins intend to address this matter promptly in an effective and thoughtful way."
Aiello said Ireland "took the proper step of calling Dez Bryant to apologize and then making that apology public."
Ireland's apology came only after Yahoo! Sports reported that he posed the question several weeks ago during a pre-draft interview with Bryant.
"My job is to find out as much information as possible about a player that I'm considering drafting," Ireland said in a statement. "Sometimes that leads to asking in-depth questions. Having said that, I talked to Dez Bryant and told him I used poor judgment in one of the questions I asked him. I certainly meant no disrespect and apologized to him.
"I appreciate his acceptance of that apology, and I told him I wished him well as he embarks on his NFL career."
Before the draft, Bryant's background received extensive scrutiny from NFL teams because of concerns about character issues. The Dallas Morning News reported last week that Bryant's mother, Angela, gave birth to him at age 15 and sold drugs. Bryant is close to his mother, who attended her son's draft-day party Thursday. He was taken by the Dallas Cowboys with the 24th pick in the first round.
NFL Players Association executive director DeMaurice Smith said Ireland's question raised worrisome issues.
"We need to make sure the men of this league are treated as businessmen," Smith said in a statement. "During interviews, our players and prospective players should never be subjected to discrimination or degradation stemming from the biases or misconceptions held by team personnel.
"NFL teams cannot have the free reign to ask questions during the interview process which can be categorized as stereotyping or which may bring a personal insult to any player as a man.
"For the past year, active, former and incoming players have heard me speak about the expectations we have of them as members of this union, their teams, communities and families. It is equally true that the same kind of respect is demanded of their employers."
The Associated Press contributed to this report.