Emma Delta has been granted an extension by Greece's privatisation agency (TAIPED) to come up with an improved bid and now has until Wednesday May 1.
The group was left as the sole suitor for OPAP on Monday when TAIPED rejected the only other bid, from New York hedge fund Third Point, for failing to meet the terms and conditions of the tender.
TAIPED's directors unanimously decided not to unseal the conditional offer from Third Point as bidders were told they could only submit definitive bids.
"Therefore, only Emma Delta’s financial offer was unsealed in the presence of its representatives," the agency said.
However, a source close to the process said Emma Delta's bid is at the bottom of the official evaluation and was not higher than €630m. Reuters reported that the consortium has submitted a €622m offer, also citing an anonymous official.
The fund is backed by Czech investor Jiri Smejc and Greek energy magnate Dimitris Melissanidis. Greek media has also speculated that Italian lottery giant Lottomatica has a stake in the Emma Delta group.
TAIPED's board of directors took the decision on Monday evening to ask for an improved offer after taking into account the recommendation of its financial advisers Deutsche Bank and NBG Securities, as well as the opinion of the agency's seven-man Council of Experts.
The board is understood to be holding out for an offer for the state's 33 percent stake in OPAP that matches the €650m price recommended by independent valuer Duff & Phelps.
The agency's board will convene next Wednesday at 14:00 (Athens time) to discuss Emma Delta's decision.
OPAP, which is one of the most valuable Greek assets up for sale, made net profit of €496m last year on turnover of almost €4bn.
Greece's government has since introduced a 30 percent tax on OPAP's gross gaming revenue which the monopoly's management team predicted will slash profit 77 percent to €116m this year.
But the rights to 35,000 video lottery terminals and a soon-to-be dominant position in online betting will help to return profits to 2012 levels by 2021, management forecasted in a long-term business plan for the privatisation.
Ministers are also adamant that a €60m dividend from OPAP will still be paid into the public purse when the deal is completed.