Correct, the requirements are only for the offer in the UK. There is no requirement for your current job.

If you are moving to the UK, there is one requirement (in the current rules) for personal savings (means thata you'll be have to have £945 in your account 90 days prior to the visa date) https://www.gov.uk/tier-2-general/eligibility. This can be traded by your employer being a premium sponsor and specifying that they can cover that requirement.