There is a 60-percent chance of good weather for a Sunday launch, according to ULA.
NASA hopes the probe will help determine which parts of the sun are providing the energy source for solar winds and solar particles, and how they accelerate to such high speeds.
If the probe doesn't launch on Sunday, the window for a successful launch doesn't close until August 23.
NASA's planned mission to explore the sun was delayed Saturday as the rocket failed to take off during the designated launch window.
The probe is created to plunge into the Sun's mysterious atmosphere, known as the corona, coming within 6.16 million kilometres of its surface during a seven-year mission.
The early morning launch countdown was halted with just one-minute, 55 seconds remaining, keeping the Delta IV rocket on its pad with the Parker Solar Probe.
"Teams worked very hard this evening, diligently getting through the launch process, looking at everything that they had to to get into the terminal count this evening", Mic Woltman, of NASA's Launch Services Program, said during NASA's broadcast of the launch attempt.More news: Jorginho scores penalty as Chelsea win and Crystal Palace spoil Fulham's return
The probe will be 3.9 million miles from the sun's surface, making it the closest spacecraft to the sun's surface in history.
The reason for the delay was not immediately clear, but was called for after a gaseous helium alarm was sounded in the last moments before liftoff, officials said.
"And it needs to be, because it takes an huge amount of energy to get to our final orbit around the Sun", Driesman added.
"The sun is full of mysteries", said Nicky Fox, project scientist at the Johns Hopkins University Applied Physics Lab.
If the teams investigate the issues that delayed Saturday's launch and it can't be resolved in time for a 24-hour turnaround, the next attempt won't happen until Monday.
The probe is set to use seven Venus flybys over almost seven years to gradually reduce its orbit around the Sun, using instruments created to image the solar wind and study electric and magnetic fields, coronal plasma and energetic particles.
The probe is protected by an ultra-powerful heat shield that is just 4.5 inches (11.43 centimeters) thick.
It is created to withstand heat of up to 1,000 degrees Celsius, speeds of 700,000 kilometres per hour and a journey that will last seven years. "Each time we fly by we get closer and closer to the Sun", Driesman added.