To bid on a player, either right-click on his name or use the Make An Offer button in the Player Profile.

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The
Transfer Offer screen will display the current valuation of the player, your remaining transfer budget and the type of offer you’re making – a full transfer, a trial (useful for signing good players from lower league clubs), a loan (borrowing the player for a period in which you pay a percentage of their wages) or an enquiry (a hush-hush chat with the other team’s manager). Co-ownership is available in some situations.
Other elements to the transfer might include Additional Clauses such as a Buy Back Price or Percentage Of Next Sale, or Additional Fees like paying in Monthly Installments or paying a lump sum After Promotion to a different league. If you’re short of cash you can also add one of your current players into the deal...
After the bid has been made (use the Make Offer button) the other club will then mull the offer over, and hopefully get back to you with a positive response.
On occasion however they may turn you down or want more money. At this stage it would be wise to consider pulling out if the deal isn’t favourable to you, but don’t be afraid of a bit of negotiation.
Once the fee is settled and an agreement has been made, the player must then decide if he wants to join your club and if he does, you’ll get the opportunity to offer a contract...