This train is a "Goin' Home" train. But it may not take American travelers home for long.

"This train passes through some of the most beautiful regions of the nation," in the words of an Amtrak brochure designed to promote the passenger train, which provides daily service along a 1,340-mile route between New York or Washington in the East and Kansas City at mid-continent.

But the National Limited is among Amtrak's chief money-losers and is considered a likely victim of a budget squeeze that is forcing the government-subsidized rail corporation to make plans for a series of cutbacks in service across the nation later this year.

Even as Amtrak's National Limited carried a nearly full train of riders between New york and Columbus, Ohio, one day earlier this week (the crowd thins out west of Columbus), President Carter was submitting his proposed budget to Congress for the next fiscal year, and the news for Amtrak was that the administration is seeking to hold the line on rail passenger deficits.

Amtrak asked for $613 million to cover deficits in fiscal 1979, and the administration has proposed $510 million, up slightly from $506.5 million in the current year. Amtrak asked for $810 million to buy new equipment and build facilities, and the administration proposed $125 million - down from $133 million this year.

Because costs continue to rise, Amtrak's management has declared that services must be cut or eliminated in the current year. If approved by Congress, the Carter budget would require additional cutbacks after Oct. 1.

The Department of Transportation is working on a study of Amtrak's routes and will recommend cutbacks within two months.

Passengers riding this week on a route that connects five state capitals and several other major cities expressed little knowledge about Amtrak's financial crisis and were not aware that the National Limited's days may be limited.

But several members of the crew showed deep pessimism and some said they thought the train hasn't been promoted or provided with adequate equipment to test its potential ridership appeal.

Indeed, with a foot or more of snow lining the train's entire route between New York and St. Louis this week, the National Limited was more often late than on time. It suffered equipment breakdowns. One attempt to restore heating to a dining car resulted in a fire.

Passengers took it all in stride for lurch, when the train traveled over particularly rough sections of Consolidated Rail Corp. track).

Kenny Young, whose train trip started in Florida and ended in Columbus after he changed trains in Washington and North Philadelphia, was one of the riders going home. In the case of Young, his wife and a 14-year old son, it was a matter of being disappointed with a short stay in the South.

The train was about two hours late into Columbus, but Young was glad to be there and had no particular complaints about Amtrak. When a generator on the old passenger car in which he was riding broke, the train was stopped for repairs.

Steven Wroten, an Ohio State University graduate student in industrial organizational psychology, was in the same car and he noted that people in other cars weren't told why there was a delay.

Wroten, a 23-year-old native of Dallas, rode the National Limited twice in three days - his first trips ever on a passenger train. He traveled from Columbus to New York late last week in the midst of blizzard and was six hours late - missing the first hour of a Radio City Music Hall jazz concert as a result.

Although the return trip also was late - forcing him to miss a scheduled teaching assignment - Wroten shared the resigned attitude of most riders. "I like talking, meeting people . . . the trip allows time to sit and think and catch up on studies," he said.

"I don't know much about the politics of Amtrak . . . but Amtrak appears to be inefficient . . . if you have time and can afford to be late, it's okay," Wroten concluded.

Despite the slow performance during recent snows, the National Limited's on-time performance has been improving. During October, 84 per cent of the National Limited's runs arrived on schedule compared with 69 per cent a year earlier - reflecting major track rehabilitation on Conrail's line west of Indianapolis.

But the National Limited is attracting fewer passengers. In the first 10 months last year, the number of riders fell by 3 percent to 155,590. More encouraging to Amtrak officials, many of the remaining riders are taking longer trips and total passengers miles rose by 17 percent.

Nevertheless, Amtrak's deficit has to be reduced and government planners likely will look at the balance sheet where they will discover heavy losses. In the last fiscal year, the National Limited's revenues were estimated at $6.3 million and the cost was $11.9 million, for a $5.6 million annual deficit.