Arnold confirmed he had been addressed by two of his Verwood players on Wednesday over rumours associating him the switch, despite Priory already having a permanent manager, Michael Cuffie, in place.
Cuffie has endured a tough opening season with the Hurn Bridge outfit currently fifth-bottom of the table on the back of six winless league matches.
Arnold turned down the chance to manage Christchurch in May, prior to Cuffie’s appointment, and was adamant he had not been approached since.
But the frustrated Verwood boss did reveal his successful eight-and-a-half year stint at Potterne Park could come to an end and cited the club’s perceived lack of ambition as the primary reason.
Arnold told the Daily Echo: “A couple of lads asked me about the Christchurch link and it was not the first I had heard about it.
“After last season’s talks, there will always be rumours about me going there and there is not a lot I can do about that.
"My frustration is over the need for backing from within the club (Verwood) and I have to consider whether they match my level of ambition.
“It is not a criticism because we have a very small committee who put in a lot of time and effort, but it needs that bit more to push on.
“The Wessex Premier will be very strong next season. Any side without a reasonable budget will struggle and I am not interested in treading water.
"I am the sort of manager who has to have targets and if we’re not raising the bar then it is not a challenge to me.
“Every target we set for the club, I have met head on and I want more. My target this season was to push on from last season’s 14th place but I have felt like I have had my hands tied.
“It makes it a much harder job when you’re standing still instead of pushing to go somewhere.
“We’ll be in the bottom six next season without a budget and while that might appeal to some, it doesn’t to me because I have confidence in my own ability to do more than that.”
In response to Arnold, Verwood chairman Steve Jefferis said: “The club fully appreciates Adie’s frustrations but despite a lot of hard work by myself and the committee, a playing budget is not achievable, certainly for the forseeable future.”
Verwood, who lost 5-0 to Alresford last night, host Newport (IOW) in the Wessex Premier on Saturday.
Christchurch will look to bounce back from a 2-1 defeat to Fareham in the Russell-Cotes Cup semi-final when they host bottom club Downton in a crunch relegation scrap.
Elsewhere, Poppies host Alresford while Hamworthy United travel to Whitchurch (all 3pm).