Bosses at the LTA are conducting a review into the embarrassing events against Lithuania, and current captain John Lloyd's future is believed to be central to that.
But BBC tennis correspondent Jonathan Overend says there is support for Lloyd to stay on.
"The players are rallying around the captain," said Overend.
"One player said on Monday night, 'There is nobody better for the job, so why get rid of him.'
"Lloyd is known to have plenty of respect in the British locker room and he will hope that counts for more than historical ignominy."
World number four Andy Murray won both his singles matches for Britain as they lost to Poland in September, but missed the match against Lithuania to concentrate on preparing for this season's Grand Slams.
The Scot, who moved to Spain as a teenager to continue his training, was supported in his decision by Draper, though David Lloyd believes Murray should have been encouraged to play.
"Murray didn't come from the system. His mother was brave enough to take him out of it," said Lloyd, who is also a successful entrepreneur with a chain of fitness and tennis clubs.
"Roger is wrong endorsing the fact that Andy shouldn't have played. That was a bad call.
"I would try and encourage him to play and give something back to the game.
"I would ask him to play against Turkey. We want him to help our young kids. We all owe something to the game."
Draper was appointed as chief executive of the LTA in 2006 after modernising Sport England, but his hiring of high-profile coach Brad Gilbert failed to galvanise the men's game.
The American stepped down from his coaching role with British number one Andy Murray after just 16 months and ended his association with the LTA altogether when his contract expired in September 2008.
David Lloyd was part of the last Great Britain team to reach the Davis Cup final in 1978, playing doubles with Mark Cox as the US won 4-1 in California.
He was sacked as Davis Cup captain in January 2000 after criticising the fitness of some of his players in the wake of a 4-1 World Group defeat by the Czech Republic.
He subsequently ran High Performance Centres with financial support from the LTA, but remained critical of the organisation.
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