Lloyd became the first GB skipper to oversee five consecutive defeats as Britain crashed to a 3-2 loss at the hands of a country with only three world-ranked singles players, all of them teenagers.
Britain now face a home relegation play-off against Turkey in July to avoid dropping into Europe/Africa Zone Group III - the lowest tier of the competition - alongside the likes of San Marino, Andorra and Montenegro.
Former Australian Open finalist Lloyd admitted yesterday he was considering his future in the role, saying: "I'll ask myself, 'Did I screw up or could someone else do a better job?"'
A newspaper report this morning suggested former British number one Rusedski, who currently works for the LTA as a talent and performance ambassador, has already been lined up to replace Lloyd.
Player director Steve Martens, though, insisted no decisions have yet been made.
He told BBC Radio Five Live: "We have had no direct contact at all (with Rusedski). I think it would also be very disrespectful to a guy like John to do that.
"Greg is a fantastic coach, he works with us, he's fully on board working with some of the male players and John so far has been fully in the loop in everything we do."
Martens continued: "Clearly we have to be swift and decisive. But it is very important that in the heat of the battle you don't do anything emotional.
"We have to be looking at this together with the captain, reviewing where we were with our players, where we were with our selection, where we were with the preparation of the players.
"That is exactly what we'll do over the next few days and the captain will be a full part of that."
LTA chief executive Roger Draper described the result as "unacceptable" and echoed Martens' words.
He said: "I share the deep disappointment and frustration at this result. Five defeats in a row is unacceptable.
"So I have asked Steven Martens to review last week's performance and result, and report back to me and the LTA main board as soon as possible.
"That review needs to be swift and decisive as it is clear some real improvements need to be made."
Rusedski, a former world number four and US Open finalist, would be an obvious choice if Lloyd does, as expected, leave the post.
The Canadian-born player represented Britain in 20 Davis Cup ties between 1995 and 2007, winning 30 of his 43 matches.
He has also been working with James Ward, Britain's number one player in Lithuania, and is known to covet the post.
Rusedski did not mince his words in his reaction to the result in Vilnius, saying on his Twitter page: "Cannot believe we have lost to Lithuania in Davis Cup. Unbelievable."
The worry for Lloyd - or his replacement - is things could conceivably get worse before they get better.
Turkey were comfortably beaten by a below-strength Ireland side over the weekend - ending the possibility of a British Isles derby.
But, in Marsel Ilhan, they have a player who is ranked inside the top 125 - substantially higher than any of the competitors on show in Lithuania - and who is on the rise.
The 22-year-old was the first Turk to win an ATP Challenger tournament and at last year's US Open he claimed another first for his country by reaching the second round of a grand slam - a feat he repeated in Australia in January.
Admittedly, below Ilhan they have little to offer - their number two player against Ireland, Haluk Akkoyun, is ranked outside the top 1,000.
But, were the tie to be taken to the pressure situation of a deciding rubber, nothing could be taken for granted.
There is surely no prospect of Andy Murray returning to the team a week after Wimbledon so the captain must again assess his slim player pickings.
Ward did himself no harm with two decent performances in Lithuania, including the first win for a Briton other than Murray, Rusedski or Tim Henman in a live singles rubber in 13 years.
Dan Evans, though, must be fearing for his place after failing to win the deciding rubber for a second consecutive match against a lower-ranked player.
Jamie Baker, who picked up an ankle injury on the eve of the weekend's match, has proved in the past he has the required fighting spirit and will certainly be considered while the likes of Josh Goodall and Chris Eaton could come into contention.
So, too, could Alex Bogdanovic. Lloyd froze the 25-year-old out of the picture after seven Davis Cup defeats in a row but, at 155 in the world, he is ranked substantially higher than any of his countrymen except Murray.
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