"We hope that some of the issues we have with the latest tomb design can be resolved amicably, and a meeting with the cathedral authorities to discuss these matters is scheduled," the society added in the statement.
The cost of the tomb and interment is estimated at 2.5 million pounds ($4.2 million), according to Monteith.
King and controversy Richard III's reburial is not the only focus on controversy surrounding the king. The University of Leicester has taken samples of DNA from the king's bones, and researchers plan to sequence Richard's entire genome. That plan has raised criticism from some, including independent historian John Ashdown-Hill.
"We're talking about a member of the royal family and a former head of state," Ashdown-Hill told Live Science in February. No other former head of state would be subject to such study, he said.
Of course, Richard III wouldn't be Richard III without controversy. The king's fame is largely due to a Shakespeare play written about him more than a century after his death. The play painted Richard as a twisted, evil villain - but the portrayal was more likely a sign of the times than based in reality. When Shakespeare wrote "Richard III," Queen Elizabeth I ruled England. Elizabeth I was the granddaughter of Henry VII, the first Tudor monarch, and the man whose forces defeated Richard III at the Battle of Bosworth Field.