The trust, which currently sits on a discount of 64% according to the Association of Investment Companies, saw its share price collapse 39.5% last month after it withdrew its 6p dividend target.
The trust's board said it had received "indicative offers from interested parties" working to secure a majority stake in Verne Global.
This sale would enable it to "accelerate its balance sheet deleveraging, deliver the cash resources necessary for the company and group to strengthen their position, and further maximise shareholder value", the board said.
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Numis analysts Andrew Rees and Ash Nandi said management was likely reluctant to sell its entire position in one of "the portfolio's marquee platforms", rather than maintain a minority stake.
Rees and Nandi added the move was not surprising "given the scale of the capital allocation challenges that DGI9 is facing, which require significant action".
Stifel analyst Sachin Saggar described the move as "unsurprising post the dividend fiasco".
He noted that a full sale of the Verne Global group, which was valued at about £517m, would "wipe out" the revolving credit facility, putting the trust in a net cash position.
"We expect, if the asset is sold in full, this will signal a process of each asset being sold individually over time. In effect, an orderly wind down of the company," Saggar said. "Arqiva will likely represent the most difficult to sell, given it is not a growth asset akin to Verne Global."
Winterflood Securities analyst Elliott Hardy said "it appears that DGI9 may live to fight another day", and the move would "likely help to soften the load" that Verne Global's growing capex requirements had placed on the trust's balance sheet.
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Hardy explained the capex requirements had totalled about £610m at the time of his last call with management.
"We view it as a positive that the decision appears to be driven by its shareholder consultation, although of course, time will tell as to whether the exit price is sufficient to help restore investor sentiment," he added.
However, Hardy noted the move raised the question as to why a full divestment was not considered sooner, "as DGI9's management failed to get a grip on actual capex requirements at an earlier stage".
"One would hope that more prudent oversight is exercised on remaining investments such as AquaComms, which management highlighted as having significant potential, to help realise value in a manner that is more accretive for DGI9 shareholders," he said.
The decision to sell a controlling stake of the business is also likely to achieve more attractive pricing for the trust, Numis' Rees and Nandi added, given the "likely preference from many potential buyers" to hold a majority position.
However, they also warned the trust may be perceived as a "forced seller", particularly in the wake of the dividend cut last month, pushing the potential price down.
Executed terms are expected to be announced later this quarter.