BBC Wales Sport understands the Gills made an enquiry for Byrne, before Edinburgh changed his mind over a player he brought to Rodney Parade.
Edinburgh opted to end the pursuit out of respect for his old club, who he managed from 2011-2015, so as not to weaken their battle against relegation.
They are currently 20th in League Two, seven points above the relegation zone.
Meanwhile, Newport have confirmed the release of defender Matt Taylor.
The 34-year old joined from Cheltenham in the summer but made just five appearances for the south Wales club.
"Further to recent discussions, we can confirm that Matt Taylor will leave the Club with immediate effect having mutually agreed to terminate his contract," a statement from the club read.
Taylor is understood to be a target for Torquay United.

Twenty-five suspects were released because of a lack of evidence.
It is the latest batch of death sentences passed in connection with the killings - 36 men were hanged a year ago for their part in the massacre.
IS filmed the killings, at the former US base of Camp Speicher, near Tikrit, in an early piece of shock propaganda.
Most of the victims are believed to have been young Shia recruits who were based at the camp when Tikrit was overrun by IS.
Photos and videos published by IS at the time showed soldiers being lined up and shot at various locations.
Some bodies were pushed into the River Tigris, while others were buried in mass graves that were found after government forces recaptured the city a year later.
The condemned men have the right to appeal against the sentences.

ViiV Healthcare, GlaxoSmithKline's HIV business, will acquire two operations specialising in late stage treatments and early phase medicines.
It is the latest acquisition in a record year for pharmaceuticals.
Global deals have reached $677.5bn, according to Dealogic, up 64% on 2014.
Healthcare companies have been forced to buy or swap assets to bolster growing areas of their businesses and shore up their defences as generic versions of previously patented medicines are released in the market.
Sales of Advair, an asthma treatment and a bestseller for GSK, are expected to decline following the introduction of a generic version in the US.
In the first nine months of the year, GSK reported that sales of Advair fell 19% to £1.2bn, whereas sales of HIV treatments rose 56% to £1.6bn.
The company had considered spinning off its 80% stake in ViiV Healthcare through a flotation that could have valued the business at £18bn.
However, it decided to hang on to the operation in May.
GSK said it would make an upfront payment of $317m to buy Bristol-Myers Squibb's late stage HIV drugs, with a further $518m dependent on sales and royalties.
The assets include Fostemsavir, which treats HIV infection and is at the final phase three development stage. That means that it can seek regulatory approval, which is expected to be filed in 2018.
It will also pay an initial $33m to acquire pre-clinical and early stage research assets and could hand over $587m if certain development and sales goals are met.
Earlier this week, AstraZeneca said it would spend $4bn to buy a 55% stake in Acerta Pharma, a US-Dutch company which is developing a leukaemia treatment that is at the phase three stage. It has an option to buy the remaining 45% in the company.
AstraZeneca, which last year saw off a £69bn takeover bid from US rival Pfizer, also announced it would acquire a number of respiratory medicines from Japan's Takeda for $575m.
This year has been marked with major deals in the pharmaceuticals sector. Last month, Pfizer announced a $160bn merger with Botox-maker Allergen while in March, GSK said it would swap its cancer drugs portfolio with Novartis' vaccines business.
Shares in GSK fell 6.5p to £13.35.
Newport midfielder Mark Byrne was a transfer target for Gillingham before a U-turn from manager Justin Edinburgh.
[NEXT_CONCEPT]
An Iraqi court has sentenced to death 27 men for their involvement in the massacre of up to 1,700 soldiers by so-called Islamic State (IS) in June 2014.
[NEXT_CONCEPT]
GlaxoSmithKline has ended a frenetic week for the pharmaceutical sector with a $1.4bn (£939m) takeover of HIV drug development assets from Bristol-Myers Squibb.