A person with knowledge of the Yankees' thinking said that reports of the discussions about a deal involving Phillips and Gardner were "way overblown."
The person said that the Yankees do not plan on trading Gardner, saying, "[If they did,] it would have to be in a valuable pitching capacity."
An MLB.com source revealed the Yankees are one of the teams listed in Phillips' limited no-trade clause. The Reds would need his approval before they could deal him to the Yankees.
New York has a void at second base after losing out on re-signing Robinson Cano, who agreed to a 10-year contract with the Mariners. The Yankees have touched base with free agent Omar Infante as a possible replacement.
The Reds, who expect to lose free agent Shin-Soo Choo, are in need of a center fielder and leadoff hitter with a short-term contract that can let them take their time to develop prospect Billy Hamilton.
Gardner, who batted .273 with a .344 on-base percentage in 145 games last season, is arbitration-eligible for a third time and can become a free agent after the 2014 season.
Phillips, 32, has four years and $50 million remaining on his contract. In 2013, he batted .261 with 18 home runs, 103 RBIs and a .310 on-base percentage.
Reds general manager Walt Jocketty told Phillips last week he was not actively shopping Phillips for a trade, but he did also say he would look at opportunities if they're presented.
Mark Sheldon is a reporter for MLB.com. Read his blog, Mark My Word, and follow him on Twitter @m_sheldon. This story was not subject to the approval of Major League Baseball or its clubs.