On Wednesday a survey showed the euro zone private sector started the second quarter on its strongest footing since 2011.
But hedge funds and speculators are betting against the euro as Draghi has warned the European Central Bank will ease policy further if the currency keeps strengthening, including the option of quantitative easing.
Other members of the ECB's Governing Council have also been vocal about the need to rein in euro strength, which has contributed to subdued inflation in the euro zone.
"The upcoming speeches by President Draghi and his Governing Council members (Klaas) Knot and (Vitor) Constancio today and tomorrow could highlight the ECB's determination to fight low inflation and, by implication, further euro appreciation," said Valentin Marinov, currency strategist at Citi.
Read MoreEuro zone business activity nears 3-year peak
"All that could keep euro bulls at bay for now."
There have been other factors behind the euro's strength, including renewed inflows into euro zone peripheral bonds and stocks; and the fact that euro zone banks are repaying cheaper loans to the ECB, shrinking its balance sheet at a time when the Federal Reserve and the Bank of Japan are expanding theirs.
Also, many euro zone banks have been cutting their presence abroad, selling assets and repatriating money to meet capital and stress test requirements. The capital inflows allow the euro zone to maintain a healthy current account balance and support the common currency.