By Ted Barrett, Kristin Wilson, Ashley Killough and Deirdre Walsh, CNN

(CNN) -- The Republican push for a broad tax overhaul faces a critical test this week as senators must approve a budget bill that would clear the way for tax reform to pass the Senate with just GOP support.

Senate Majority Leader Mitch McConnell is under tremendous pressure from his right flank -- inside and outside the Capitol -- to get the votes to advance the tax bill. After the double failure of health care reform, senators have few major legislative accomplishments to point to and Republicans -- especially President Donald Trump -- are getting antsy for some wins.

If at least 50 of the 52 Senate Republicans vote for the budget bill later this week, it will be an early victory for the process, though that won't assure the tax package, which is still under intense negotiations, can pass sometime in the next few months.

At least four Republicans senators are undecided on the bill, citing various concerns: Susan Collins of Maine, John McCain of Arizona, Rand Paul of Kentucky and Bob Corker of Tennessee.

Corker, who is not seeking re-election next year, is concerned his party is forgetting its deficit-cutting roots. The member of Senate budget committee is pushing to offset the costs through closing loopholes and getting rid of major deductions currently allowed in the tax code.

"You cannot do permanent tax reform without loophole closing," Corker told reporters recently comparing lawmakers' efforts to cut back on popular deductions to trying to get kids to choose between vegetables and dessert.

"I don't see any spinach," Corker said. "All I see if a bunch of sugar thrown on the table. No one wants to eat even one leaf of spinach yet."

Sen. Thad Cochran, R-Mississippi, who has been ill, is expected to be back to work this week in time to cast a vote for the budget, his office said.

A final vote on the budget is not scheduled yet but it will follow what could be a multi-hour "vote-a-rama" when senators can offer an endless number of amendments to the budget.

Everything else: Disaster aid, Loretta Lynch and Callista Gingrich

Also this week, the Senate is expected to vote on a nearly $36.5 billion emergency funding bill to respond to the spate of damaging hurricanes and wildfires that have hit the US recently. The House passed the bill last week despite concern from some conservative Republicans who wanted to offset the costs with cuts elsewhere.

On the Russia probe, former Attorney General Loretta Lynch is expected to meet privately with investigators from the House and Senate intelligence committees and the Senate judiciary committee.

Senators have been away for a week on recess. They will return to find their Twitter feeds filled with new policy directives from Trump on health care, the Iran nuclear deal, immigration policy and more.

On Monday evening, the Senate is scheduled vote on Callista Gingrich to be ambassador to the Vatican.

The House is out this week but Speaker Paul Ryan continues to tout the GOP's plans to overhaul the tax code. He will speak at a lunch in Madison, Wisconsin on Monday to a group of business leaders.

The Senate judiciary committee will hear Wednesday from Attorney General Jeff Sessions as part of the panel's oversight of the Justice Department.

The Homeland security committee will also hold a hearing Wednesday to consider the nominations of Jeff Pon to be director of the Office of Personnel Management and Emily Murphy to be the administrator of the General Services Administration.