The Celta Vigo striker had a medical at Liverpool's Melwood training ground on Tuesday, but flew back to Spain the following day with the transfer yet to be completed.
Liverpool are thought to have settled on a fee of £7million for Aspas, whose 12 goals in La Liga last season helped his club avoid relegation.
Jonathan Aspas, the striker's brother and agent, told Spanish radio station Radio Galega he was confident that the paperwork would go through without a hitch.
He said: "There should be no problems. Now only bureaucratic issues remain unresolved."