The post has become vacant after Lars Lagerback stepped down last week and Swedish FA president Lars-Ake Lagrell has identified Eriksson, who is now director of football at Notts County, as his replacement.
Lagrell told BBC Sport: "I am a friend of 'Svennis' and know how good a manager he is."
It is understood the Swedish FA will also ask Notts County for permission to speak to the 61-year-old former England manager.
It is by no means certain that Eriksson would take the Sweden job even if he was offered it.
Eriksson has a five-year contract with Notts County and according to sources close to the Swede he is very keen on seeing the project through to the end.
If he was interested in the Sweden post, it is understood he would first explore opportunities to see whether he could combine both the international job with his club one.
Eriksson's business manager Lars Sternmarker told Swedish newspaper Aftonbladet: "Lars-Ake has spoken to both me and Svennis although I cannot reveal what was said for confidentiality reasons.
"But Svennis is Swedish at heart and feels very happy and flattered."
Asked whether he thought Eriksson would take the job, Sternmarker added: "The federation has identified a number of options, and the fact they have called Svennis is not particularly strange since he is the most successful coach in Sweden.
"The purpose of these calls is to allow the federation to look over the possibilities, options and what the person himself wants.
"We cannot give any definite answers for a while, we think it's fun that a dialogue has been opened and that there is scope for more discussions."