Mr Moore has sent a fundraising email to supporters asking for contributions to his “election integrity fund’ so he could investigate reports of voter fraud.
“I also wanted to let you know that this battle is NOT OVER!” he wrote.
The former judge was defeated by Democrat Doug Jones by about 20,000 votes earlier this week. It is the first time a Democrat has held the seat since 1992.
Under Alabama law, a recount is only triggered if a winning candidate’s lead is less than 0.5 per cent.
But Mr Moore, whose campaign dogged by allegations of sexual misconduct with teenage girls during the 1970s, is yet to concede the race to fill the seat previously held by US Attorney General Jeff Sessions.
He told supporters the vote was “close” and some military and provisional ballots were not expected to be counted until next week.
Mr Moore said his campaign was collecting “numerous reported cases of voter fraud” to send to the secretary of state’s office.
Secretary of State John Merrill has nonetheless said it is unlikely last-minute ballots will change the outcome of the election or even trigger a recount.
He added his office had investigated claims of voting irregularities but “had not discovered any that have been proven factual in nature.”
Mr Trump, who had endorsed the Republican candidate, called Mr Jones to congratulate him on his victory and said on Friday said that he believed Mr Moore should concede.
The Alabama Republican Party has also urged its supporters to accept the result, which has seen the state elect its first Democratic senator since 1992.
The final results of the race are expected to be certified between December 26 and January 3 after counties report official totals.
Additional reporting by Associated Press