Illinois and Tennessee are trading potshots in a fracas over the hunting limits on Canada geese, and the Interior Department's Fish and Wildlife Service appears to be caught in the cross fire.

At issue are the federal government's proposed 1982 targets on the number of geese that hunters will be permitted to bag in the Mississippi Valley flyway, a 14-state area that extends from Canada to the Gulf states.

The migratory goose population in the flyway has been declining in recent years, and the Fish and Wildlife Service -- with what it thought was the agreement of The States involved -- embarked on a management plan to bring the numbers up.

But when Illinois officials found out that the plan involved cutting by 10 percent the number of geese their hunters could take, they cried foul. And when word reached Tennessee that Illinois was pressing to increase its quota, there was quite a flap. In Tennessee, the hunting season on Mississippi flyway geese has been closed for years to let the population build up.

Both sides migrated to Washington this week to lay their problems at the feet of Assistant Interior Secretary G. Ray Arnett in separate meetings arranged and attended by their congressional allies.

The Fish and Wildlife Service wants to cut Illinois from the 30,000 birds of last year to 27,000 this year. "It's going to reduce the number of birds to be harvested and make our hunters extremely unhappy--we're not going to be very popular, I can tell you," said Illinois conservation official Pete Miller after the first audience Wednesday.

But Tennesseans charge, and Miller concedes, that Illinois has been less than successful in the past in adhering to the federal quotas. Miller said hunters in the state shot about 44,000 geese last year, 14,000 more than the federal limit for the state. But he contends that the state has made "stringent changes" to reduce that take to 30,000 this year, and there is no need for the federal government to further reduce the number.

The southerners, who got the chance to argue their point with Arnett yesterday, disagree. Jim Range, an aide to Sen. Howard H. Baker Jr. (R-Tenn.), said the Illinois officials "understand that goose-hunting is extremely lucrative."

Range's point is that most of the geese in Illinois are taken on land owned by hunting clubs that charge fees. Dr. John Rogers, head of the migratory bird office at FWS, acknowledged that Illinois had argued the new limits "would be an economic loss to them."

Tennessee, however, is decidedly unwilling to curb its hunters for the sake of Illinois pocketbooks.

Somewhere in the middle of all this is the Fish and Wildlife Service, which set the new limits only after the Mississippi Flyway Council, composed of representatives of all 14 affected states, could not agree on the number of geese that could be taken without adversely affecting the population. The northern states wanted a limit of 100,000 for the 14-state flyway, the southern states wanted 40,000, and FWS settled on a compromise figure of 65,000.

The regulations are scheduled to be published in the Federal Register Monday, and FWS spokesman Alan Levitt said Arnett was unlikely to make changes in the quotas until the comment period had passed, if he makes any at all.