It's still a bitterly fought proxy battle, but might Hess (NYSE:HES) and activist shareholder Elliott Management be moving toward conciliation?

The two sides have been trading sharp barbs for a year now, and yesterday the oil company lambasted the hedge fund operator's board nominees for having "compromised their independence and judgment" for agreeing to receive payment from Elliott. It's offered the shareholder two seats on the board, but Elliott has characterized the move as a "P.R. stunt."

Today, however, the hedge fund has offered up a counter-proposal suggesting it get five board members seated alongside Hess' five board members in a reconstituted board. Perhaps there's a thaw in the air.

Last week Hess said it would agree to split the role of chairman and CEO, one of the demands Elliott has made, though the hedge fund said it was only done after realizing the tide was turning against the oil company. Similar sniping has marked the proxy battle since the beginning.

With the shareholder meeting just days away on May 16 and the rhetoric hot on both sides, perhaps there's a cooling-down period coming that will be reflected in an agreement before investors have to weigh in.

The investment firm said of its latest proposal to reconstitute the board: "Shareholders want real change and a renewed board. Hess has promised such renewal, and this solution will follow through on that promise."

Whether the oil company agrees remains to be seen, as it has not yet responded to the latest offer.

Fool contributor Rich Duprey and The Motley Fool have no position in any of the stocks mentioned. Try any of our Foolish newsletter services free for 30 days. We Fools don't all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. The Motley Fool has a disclosure policy.