Either they will give him the nod to double the size of Farnell with an ambitious US takeover, letting the component distributor rub shoulders with Britain's biggest companies, or they will block the deal and his days as chief executive will be numbered.
In the balance is the proposed acquisition by Farnell of Premier Industrial, which dominates the unglamorous but extremely lucrative American market for catalogue sales of electrical components. A large deal, worth pounds 1.85bn, it has aroused unprecedented dissent from normally docile fund managers, some of whom see the deal as risky, vain and, most importantly, overpriced.
Unusually, institutions such as Standard Life and Legal & General have abandoned the traditional behind-closed-doors approach to corporate governance and attempted to drum up support for a no-vote by making an early, public statement of their opposition and garnering the unequivocal support of the Financial Times. With Farnell needing approval from 75 per cent of the votes cast this week, Thursday's ballot is now too close to call despite having votes in the bag from Mercury Asset Management, Scottish Widows and Flemings, who between them control 23 per cent of the shares.
Mr Poulson, a Lancastrian running one of Yorkshire's most successful companies, is used to playing the outsider but he seems genuinely bemused by the City's high-handed reaction to what he sees as the opportunity of a lifetime for Farnell: "We always assumed a nervous knee-jerk reaction because it is a large deal and we had no way of preconditioning the City to our thinking. But no one has questioned the industrial logic."
Farnell's supporters, in fact, believe the acquisition of Premier will be the company's defining deal, establishing the company as a global player in the increasingly international component distribution industry. "You can't be a small UK distributor and hope to survive in the world electronics market. You have to be a world player" is the message Poulson has been repeating more and more hoarsely through 60 or so separate presentations over the past three weeks.
But Standard Life, Legal & General and, according to one weekend report, now the influential Prudential as well will cast their votes against the deal on Thursday. At three-and-a-half times Premier's turnover, they say, the price Farnell is paying shows it has taken its eye off the financial ball by focusing too much on the compelling industrial sense of the merger.
Less than a week before the crucial poll, Poulson chooses his words carefully, but he plainly believes that is so much tosh: "I'm not going to dash in and pay over the odds just because it's there. The exit p/e is about 24. That premium has to be compared not with Farnell but with Electrocomponents (a rival distributor) which is the same kind of business and trades on 21 times prospective earnings. Do you think you could buy Electrocomponents on that multiple?"
Glad-handing fund managers is not second nature to Poulson, variously described as having "a will of steel" and being "autocratic". Driving around Farnell's home town of Wetherby in his other passion, a red E-type Jaguar, he obviously cares little for what others may think of him.
Is he a business tyrant? "Well, let's just say I don't find it difficult to decide where we should go. I don't think you should sit around all night debating issues. I'm not frightened of making decisions. I don't like politics and I don't like time-wasters."
By effectively saying that they don't think Farnell's management is up to running a global business that within months of the deal could be knocking on the door of the FT-SE 100, the dissenting institutions have turned Thursday's ballot into a vote of no-confidence.
As one supportive investor said last week: "Some fund managers are developing an overblown sense of their own importance. We are as concerned as anyone else, but one of the things that worries us is that if the deal is voted down, what happens to the management? They would have to resign. Is that really what shareholders want?"
It surely is not. Poulson's no-nonsense approach to growing Farnell has made him enemies, not least at the manufacturing arm that was an early victim of the focus on the company's core distribution competence. But shareholders, who have seen their investment grow five-fold in as many years, have nothing to complain about.
Poulson refuses to take it personally: "Look, I'm not on any kind of ego trip. People have got the size thing out of all proportion. Actually the two companies are about the same size. In no sense am I underestimating the management task, but this would be one hell of a gain for the company."
The key to Thursday's outcome will be the number of shareholders Farnell can persuade to vote - the bigger the total number of votes cast, the more supporters the rebels will have to find to shoot the deal down. Standard Life only accounts for 2 per cent of Farnell and Legal & General even less.
They are counting on as few as 40 per cent of the eligible votes being cast, which would mean opposition from holders of 10 per cent of the equity would be enough to scupper the bid. With 55 per cent of shareholders having had their say at a recent, uncontentious annual meeting, however, that seems improbably low and the odds still favour approval.
As he enters the final straight, Mr Poulson remains resolutely optimistic: "It would be a great shame if the deal was voted down. It is very clear that the overwhelming majority are in favour."
"Over the last 10 days I have spent a lot of time talking to shareholders, going over the detail, and the the mood has changed quite dramatically. There is a growing realisation that this is a beneficial deal."