I am a junior software developer thus it's not very profitable for a company on their own and I fully understand it. But I always wanted to live abroad.
I do not need any visas (I am an EU citizen) and I will apply only in a position in the EU. How should I write about it in order to this message does not sound like a scam?
Should I just write 'I can relocate as soon as possible at my own.' Does it sound credible?
There is no need to mention this in your resume or cover letter. Having to pay yourself for your relocation (if any), is the default anyway.
However, if a company is willing to pay relocation money, it would be foolish to throw that away by stating beforehand you don't need it. If the company feels it's something which needs to be cleared up during the interview process, they will bring it up.
While other answers are correct in saying you do not need to mention it and if the company is paying, you should just accept it, they may be missing your main reason behind this.
You may be trying to make yourself more "attractive" to the company (or less of "burden") during the application review phase itself. If they toss out your resume simply because you are far away, you lose an opportunity for the reasons which is not a deal-breaker for you.
If that is the case, you can simply add a line
Willing to relocate anywhere in EU at a short notice.
Though not explicitly mentioned, this kind of indicates that relocation money is not your prime concern.
You could mention in your cover letter that you'd love to live in the city the company is based. Although it isn't necessary, the fact that you're applying there already suggests you'd be ready to move there.
But no, don't say you're ready to pay for the relocation yourself, unless they say they won't be able to pay relocation money. Or are only to pay a limited amount. It's common for many companies to pay relocation fees, it isn't that big of a deal for them. If they have limitations, they will communicate them to you.
If company is willingly paying for relocation, It would be a waste to let it go. If you are specifically considering cost for visa applications, You can mention that you are citizen of X country and would not require any sponsorship for visas. You can mention safely mention this in your cover letter as
I hold valid passport of country X which allows me to legally work at location Y and will not require any visa sponsorship.
