Edward S. Lampert needs to up his offer if he wants to stay in the game and keep Sears Holdings Corp. as a going concern comprised of 425 stores.

According to the bankruptcy court’s docket sheet, a status hearing is set for Tuesday at 10 a.m., which means that a decision on the future of Sears could come later that day. Currently, it is believed that the $4.4 billion offer by Lampert — Sears’ chairman, as well as the chairman of hedge fund ESL Investments and chief executive officer of ESL affiliate Transform Holdco LLC, which is making the bid for Sears — didn’t provide enough funds to take care of administrative claims and costs in the bankruptcy.

The issue of having enough cash to fund the bankruptcy isn’t a new one for the retailer. Late last year when the company sought approval for additional incremental debtor-in-possession financing, the committee of unsecured creditors in court filings had argued against the new financing.Their position was centered on the cash burn rate at Sears and how letting the company operate while it tried to find a buyer could mean that it might run out of cash around February, and then be administratively insolvent.

In the meantime, discussions were said to be ongoing throughout the weekend to see if there could be some resolution to the shortfall issues. Lambert doesn’t tip his hand, ever, so the initial offer was likely a lowball bid to see what offers were coming in from liquidators, which are believed to be higher than Lampert’s $4.4 billion. The question is how much higher Lampert would be willing to go to match those numbers.

While Lampert has a back-up plan in place to bid individually for certain assets, he is expected to at least try to negotiate terms that would be sufficient to get him to the Jan. 14 auction. His incentive to getting a deal done that satisfies what the constituency groups want has to do with obtaining releases to forestall any future litigation connected to some of Sears’ past deals. Unsecured creditors have raised issues connected with the transactions. And because those deals were done with Sears’ board approval, not to mention the advice of financial and legal advisers, it isn’t clear whether anything wrong can be found.

But in the Sears bankruptcy, nothing’s certain. And one other sticking point is Lampert’s $1.3 billion credit bid component, which has met with resistance from the unsecured creditors. While allowed under federal bankruptcy law, they are opposing it because they want Lampert to put more cash into the deal.

Sears filed a voluntary Chapter 11 petition for bankruptcy court protection on Oct. 15. It operates under the nameplates Sears and Kmart.