Both United and US Airways declined to comment.
The two carriers have been in talks about a possible merger for a few months, while United was also in talks with Continental Airlines for a full merger.
But United's merger talks with US Airways picked up steam in late April, after Continental called off full-merger talks with United.
The wave of talks come after Delta Air Lines and Northwest Airlines said in April they planned to merge and become the world's
largest airline, seeking to counter skyrocketing fuel prices, a weak economy and growing competition from European carriers as trade barriers fall on trans-Atlantic travel.
After racking up $35 billion in losses and finally emerging from a 5-year slump in 2006, U.S. airlines are hoping mergers or alliances could give them greater market power to reduce flights and raise fares.
The airlines also face a renewed sense of urgency to cut costs as jet fuel prices have more than doubled since the start of last year.
Both Tilton and Parker have been proponents of industry consolidation but the pace of merger talks between United and US Airways had slowed lately over concern about how they could raise capital to fund the integration of the two carriers, the source said.
At Thursday's meeting, Tilton also expressed concern about United's pilots' opposition to the merger, the source said.
United's unions had said they would oppose a merger with US Airways. Pilots at US Airways were also wary of a potential deal with United, and has said United's "financial health is a major concern."
UAL lost $537 million in the first quarter. Wall Street analysts are concern it might not be able to comply with bank covenant and loan agreements, though the carrier has said it faces no such problems.
Airline mergers and the integration of fleets, labor and infrastructure can be very expensive and time-consuming.
Integrating two labor forces could take years unless the two unions agree to terms before a merger is announced.
US Airways, which merged with America West in 2005, still operates its staff with two separate labor contracts and has had trouble integrating its customer reservation systems.
United's Tilton told Parker on Thursday the carrier is close to forming an alliance with Continental Airlines with plans to seek antitrust immunity. Continental, meanwhile, is also having similar talks with AMR's American Airlines and British Airways, other sources have said.
Airline alliances allow partners to streamline costs while sharing revenues. Without antitrust immunity, the data and revenue shared on the routes would normally be considered collusive.